Extremal Combinatorics in Generalized Kneser Graphs

Extremal combinatorics in generalized Kneser graphs

Citation for published version (APA): Mussche, T. J. J. (2009). Extremal combinatorics in generalized Kneser graphs. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR642440

DOI: 10.6100/IR642440

Document status and date: Published: 01/01/2009

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Download date: 04. Oct. 2021 Extremal combinatorics in generalized Kneser graphs

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magniﬁcus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 16 april 2009 om 16.00 uur

door

Tim Joris Jacqueline Mussche

geboren te Eeklo, Belgi¨e Dit proefschrift is goedgekeurd door de promotoren: prof.dr. A.E. Brouwer en prof.dr. A.M. Cohen

Copromotor: dr. A. Blokhuis Contents

Contents 3

1 Introduction 7 1.1 Sets ...... 7 1.2 Graphs ...... 9 1.2.1 Basic deﬁnitions ...... 9 1.2.2 Graph colorings and chromatic numbers ...... 11 1.2.3 Graph homomorphisms ...... 13 1.3 Finite projective spaces ...... 15 1.3.1 Finite ﬁelds ...... 15 1.3.2 Projective spaces ...... 17 1.3.3 The projective space PG(n, q) ...... 22 1.3.4 Some counting in PG(n, q) ...... 24 1.4 Finite polar spaces ...... 25

2 Combinatorics and q-analogues 27 2.1 An easy example ...... 27 2.2 Sperner’s Theorem ...... 28 2.2.1 Original problem ...... 28 2.2.2 q-analogue ...... 30 2.3 Erd˝os-Ko-Radotheorem ...... 31 2.3.1 Original problem ...... 31 2.3.2 q-Analogue ...... 32 2.4 Bollob´as’stheorem ...... 33 2.4.1 Set version ...... 33 2.4.2 Set vs. subspace version ...... 36 2.4.3 Subspace version ...... 36 2.5 The Hilton-Milner theorem ...... 36 2.5.1 Original problem ...... 36

3 4 CONTENTS

2.5.2 The q-analogue ...... 38 2.6 Small maximal cliques ...... 53

3 The Kneser and q-Kneser graphs 59 3.1 Deﬁnitions and properties ...... 59 3.2 Homomorphisms...... 61 3.2.1 Homomorphisms between Kneser graphs ...... 62 3.2.2 Homomorphisms between q-Kneser graphs ...... 64 3.2.3 A homomorphism from a q-Kneser graph into a Kneser graph ...... 65 3.3 Chromatic numbers ...... 65 3.3.1 The Kneser graphs ...... 65 3.3.2 The q-Kneser graphs ...... 68

4 A family of point-hyperplane graphs 77 4.1 Deﬁnition ...... 77 4.2 PH(2, q)...... 83 4.3 PH(3, q)...... 83

5 Polar versions of the q-Kneser graphs 87 5.1 Deﬁnition ...... 87 5.2 Chromatic numbers ...... 89 Q+ 5.2.1 Kq (2m + 2, m + 1), m ≥ 2 even, a trivial case . . . . 89 P 5.2.2 Kq (n, 1) ...... 90 P P 5.2.3 Γq (n, 2) and Kq (n, 2), where P has rank 2 ...... 94

6 Generalized Kneser graphs 99 6.1 Chevalley groups ...... 99 6.1.1 Tits systems ...... 99 6.1.2 Coxeter systems ...... 101 6.1.3 Root systems ...... 101 6.1.4 Chevalley groups ...... 103 6.2 Generalized Kneser graphs ...... 105 6.2.1 Deﬁnition ...... 105 6.2.2 Parameters ...... 105

6.3 Chromatic numbers in the thin An(1) case ...... 114 6.4 Chromatic numbers in the thick An case ...... 121 CONTENTS 5

A Parameters of generalized Kneser graphs 123 A.1 The case J w0 = J ...... 123 A.2 The case J w0 6= J ...... 143

Bibliography 148

Index 153

Samenvatting 157

Summary 159

Acknowledgements 161 6 CONTENTS Chapter 1

Introduction

In this ﬁrst chapter we will introduce the basic notions used in this thesis. It is not the intention to give a complete, self contained description of every topic. Instead the most important notions are deﬁned and the most useful results are mentioned, often without proof. We refer the reader to the ci- tations with each topic for more background information and proofs of the results.

First we introduce some notions of set theory. Then some graph theoretic topics are mentioned. Since this thesis describes a way of translating problems in set theory and graph theory into ﬁnite geometry we need to introduce the basic facts about projective spaces. In Chapter 5 we generalize the construction over projective spaces of Chapter 3 to work over polar spaces, so we deﬁne here what polar spaces are and give some properties.

Unless stated otherwise all the objects used here are ﬁnite.

1.1 Sets

Take a set X with n elements. Such a set is also called an n-set. We can ask ourselves how many subsets of X with k elements (k-subsets) there are, where 0 ≤ k ≤ n. This is of course a very easy calculation, but in a later section we will use the same method to calculate some numbers about vector spaces, so we will give the calculation anyway. To form such a k-subset we have to choose k elements from X. For the ﬁrst element we have n choices, for the second element n − 1 choices, and so on. Finally for the k-th element

7 8 CHAPTER 1. INTRODUCTION we have n − k + 1 choices left. But now there are many choices that lead to the same k-subset; indeed all choices of the same k points but in a diﬀerent order give rise to the same subset. So, to get the correct number we have to divide by the number of permutations of k points, which is k!. So we have:

n(n − 1)(n − 2) ··· (n − k + 1) (the number of k-subsets of an n-set) = k! n! = k!(n − k)! n = k

Deﬁnition 1.1. The number of k-subsets of an n-set (with 1 ≤ k ≤ n) is n given by the binomial coeﬃcient k .

Deﬁnition 1.2. A set of sets F is called a set system or a family of sets. If all the elements of F are subsets of some set X, then F is called a family over X, and X is called the universe of F. If all the members of F have size k, then F is called a k-uniform family. F is called uniform if it is k-uniform for some integer k.

We give two examples of special set systems.

Deﬁnition 1.3. A set system F is called a chain when for any two elements one is a proper subset of the other.

Deﬁnition 1.4. A family of sets F is called an antichain if no set in F is a proper subset of another set in F. Easy examples of such families are uniform families. Antichains are also named Sperner families after E. Sperner, who proved a tight upper bound on the size of an antichain (see Theorem 2.2).

A set system can be viewed as a poset (or partially ordered set). To deﬁne this notion we need to deﬁne a partial order. 1.2. GRAPHS 9

Deﬁnition 1.5. A relation on a set X is called a partial order on X if it satisﬁes:

• Reﬂexivity: x x for all x ∈ X,

• Antisymmetry: if x y and y x then x = y for all x, y ∈ X, and

• Transitivity: if x y and y z then x z for all x, y, z ∈ X.

If x y or y x then x and y are called comparable with respect to , otherwise they are called incomparable.

Deﬁnition 1.6. A poset (or partially ordered set) is an ordered pair (X, ) where X is a set, called the ground set, and a partial order on X.

A chain in a poset is a subset of X whose elements are pairwise comparable. An antichain is a subset whose elements are pairwise incomparable. A maximal chain is a chain that cannot be extended to a larger chain.

It is clear to see that the relation ⊆ (“is subset of”) is a partial order. So the pair (F, ⊆), where F is a set system is a poset. It is easy to see that a(n) (anti-)chain in this poset is exactly what we called a(n) (anti-)chain before. The poset (2X , ⊆), where 2X is the set system containing all subsets of X, is called the subset poset of X.

1.2 Graphs

1.2.1 Basic deﬁnitions Deﬁnition 1.7. A (simple) graph Γ = (V,E) is a pair consisting of a set V of vertices and a 2-uniform family E, called the edge set, with universe V , whose members are called edges. Two vertices u, v ∈ V are called adjacent if {u, v} ∈ E and this edge is said to join the vertices u and v. In that case, u and v are also called the end points of the edge {u, v}. The neighbors of a vertex are the vertices adjacent to that vertex. The degree of a vertex is the number of its neighbors. A graph in which each vertex has the same degree is called regular.

We will sometimes use |Γ| as a notation for the number of vertices of Γ. 10 CHAPTER 1. INTRODUCTION

Deﬁnition 1.8. Two vertices u, v ∈ V are called joined if there are vertices u = u0, u1, . . . , um = v in V for a certain m ≥ 0 such that {ui, ui+1} is an edge for all 0 ≤ i ≤ m − 1. Such a sequence of adjacent vertices is called a path between u and v. If all the ui’s are diﬀerent, this is called a simple path. If u and v are connected, the length of the shortest path between them is called the distance and is denoted by d(u, v). A path of length at least three without repeated vertices except that the two endpoints are the same is called a cycle.

Being joined by a path is an equivalence relation on the set of vertices. The equivalence classes are called the connected components of the graph. A graph is called connected when it has precisely one connected component. Thus the empty graph (the graph with no vertices) is disconnected because it has no connected components.

Some subsets of the vertex set have special properties:

Deﬁnition 1.9. An independent set is a subset of the vertex set in which no two vertices are adjacent. A clique is a subset of the vertex set in which all pairs of vertices are adjacent. A complete graph is a graph where the edge set E contains all pairs of vertices in V . This means that all vertices are pairwise adjacent. The complete graph on n vertices is denoted by Kn.

Starting from one graph, there are many ways to construct other graphs. The most straightforward ways are taking subgraphs and taking the complement:

Deﬁnition 1.10. A graph Γ0 = (V 0,E0) is called a subgraph of a graph Γ = (V,E) if V 0 is a subset of V and all edges in E0 are also in E. A subgraph is called induced if all unordered vertex pairs of Γ0 that are adjacent in Γ are also adjacent in Γ0.

Deﬁnition 1.11. The complement of a graph Γ = (V,E) is the graph Γ = (V, E) where the edge set E consists of all pairs of vertices that are not elements of E. In other words, adjacent pairs in Γ are not adjacent in Γ and vice versa.

It is clear that the complement of a clique is an independent set and that the complement of an independent set is a clique. For that reason, an independent set is also called a coclique. The complement of the complete 1.2. GRAPHS 11 graph Kn has n vertices and no edges and is called an edgeless graph.

To each graph we can associate certain numbers, called graph parameters. Some well-known examples are the independence number and the clique number.

Deﬁnition 1.12. The independence number α(Γ) of a graph Γ is the size of the largest independent set in Γ. The clique number ω(Γ) is the size of the largest clique in Γ. Again it is clear that α(Γ) = ω(Γ) and ω(Γ) = α(Γ).

1.2.2 Graph colorings and chromatic numbers Definition 1.13 (Graph coloring). A vertex coloring of a graph Γ = (V,E) is a partition of the vertex set. The color of a vertex is determined by the block of the partition that the vertex is in. A proper (vertex) coloring is a vertex coloring such that the end points of an edge have a different color. From now on, if the term (graph) coloring is used without further qualifi- cation, we are referring to a proper vertex coloring of a graph. Colors will usually be denoted by integers.

We can now deﬁne some other graph parameters:

Deﬁnition 1.14. The chromatic number χ(Γ) of a graph Γ is the minimum number of colors needed to color Γ. A coloring using only χ(Γ) colors is called a minimal coloring of Γ.

Because all vertices of the same color form an independent set, there is a very easy connection between the independence number and the chromatic number of a graph:

Proposition 1.15. For each graph Γ we have that

|Γ| ≤ α(Γ)χ(Γ).

Graph colorings are generalized by multiple graph colorings: 12 CHAPTER 1. INTRODUCTION

Deﬁnition 1.16 (Multiple coloring). A k-fold coloring of a graph Γ for a positive integer k is an assignment of exactly k colors to each vertex of Γ such that two adjacent vertices have no colors in common.

A connected graph Γ with at least two vertices must have at least one edge. In a k-fold coloring of Γ with c colors, the endpoints of this edge cannot have a color in common, so we have:

Proposition 1.17. If a k-fold coloring of a connected graph Γ exists with c colors, then:

• c = mk if Γ is an m-clique,

• c ≥ mk if Γ contains an m-clique.

Associated with these multiple colorings is the multiple chromatic number:

Deﬁnition 1.18. The k-fold chromatic number χk(Γ) of a graph Γ for a positive integer k is the minimum number of colors needed for a k-fold coloring of Γ. Such a coloring is called a minimal k-fold coloring of Γ.

It is obvious that a 1-fold coloring of a graph is just a proper vertex coloring.

It is easy to see that χk(Γ) is subadditive:

Proposition 1.19. For each graph Γ and positive integers k1, k2 we have:

χk1+k2 (Γ) ≤ χk1 (Γ) + χk2 (Γ).

Proof. Let C1 be a minimal k1-fold coloring of Γ and C2 a minimal k2-fold coloring such that C1 and C2 have no colors in common. Now each vertex of Γ is colored by exactly k1 colors of C1 and exactly k2 colors of C2, so we have a (k1 + k2)-fold coloring of Γ and hence an upper bound on χk1+k2 (Γ). Deﬁnition 1.20 (Fractional chromatic number). The fractional chromatic number χF (Γ) of a graph Γ is deﬁned as:

χk(Γ) χF (Γ) = lim k→∞ k 1.2. GRAPHS 13

The existence of this limit is guaranteed by the previous proposition and the following lemma by M. Fekete:

Lemma 1.21 (M. Fekete (1923) [20]). If a sequence of real numbers {an} satisﬁes the subadditivity condition, then a a lim n = inf n . n→∞ n n n

For the fractional chromatic number of a graph, this means:

χk(Γ) χF (Γ) = inf . k k

We state some properties regarding fractional chromatic numbers:

Proposition 1.22 ([43]). Given a graph Γ we have that:

(i) χF (Γ) is a rational number,

χk(Γ) (ii) there is a positive integer k such that χF (Γ) = k , and

|Γ| 1 (iii) χF (Γ) ≥ α(Γ) , with equality if Γ is vertex-transitive .

Note that the second property means that the fractional chromatic num- χk(Γ) ber of a graph Γ is actually the minimum of all k .

1.2.3 Graph homomorphisms Homomorphisms from a graph Γ into certain other graphs can give information about the parameters of Γ. We will give the deﬁnition of a graph homomorphism and show what “target graphs” give information about the various chromatic numbers.

1A graph is vertex-transitive if the automorphism group of the graph is transitive on its vertices 14 CHAPTER 1. INTRODUCTION

Deﬁnition 1.23 (Graph Homomorphism). Consider the graphs Γ = (V,E) and Γ0 = (V 0,E0). A (graph) homomorphism η :Γ → Γ0 is a map from V to V 0 such that adjacent vertices of Γ are mapped to adjacent vertices of Γ0. This implies that the ﬁbers2 over the vertices in Γ0 are independent sets of Γ.

If the map is injective, we call the homomorphism an embedding.A surjective embedding whose inverse is also a homomorphism is called an isomorphism. Two graphs that have an isomorphism between them are called isomorphic.

If there is a homomorphism from Γ to Γ0 we will note this as follows: Γ → Γ0. If there is an embedding from Γ into Γ0 the notation becomes: Γ ,→ Γ0. We will write Γ =∼ Γ0 if Γ is isomorphic to Γ0.

Graph homomorphisms can be used to obtain bounds on the various chromatic numbers of a graph. Suppose there is a homomorphism η :Γ → Γ0, and take a (k-fold) coloring of Γ0 (for a positive integer k). Now give the vertices of Γ the same color(s) as their images under η. It is clear that this yields a proper (k-fold) coloring of Γ. This shows the following proposition: 0 0 Proposition 1.24. If Γ → Γ , then χk(Γ) ≤ χk(Γ ) for all positive integers k.

A coloring of a graph Γ with c colors can be seen as a homomorphism Γ → Kc. Indeed, number the colors m1, m2, . . . , mc and the vertices of Kc v1, v2, . . . , vc, and deﬁne for all vertices x of Γ:

η(x) = vi where mi is the color of x. Two adjacent vertices of Γ must have a different color, so they will be mapped to different vertices of Kc, and those are adjacent, so η defines a homomorphism.

Conversely every homomorphism of a graph into a complete graph can be seen as a coloring of the graph with as many colors as vertices of the complete graph. So we have: Proposition 1.25. A coloring of a graph Γ with c colors is equivalent to a homomorphism from Γ to the complete graph Kc.

2inverse images 1.3. FINITE PROJECTIVE SPACES 15

With this in mind we can give an alternative deﬁnition of the chromatic number of a graph:

Proposition 1.26. χ(Γ) = min{c such that Γ → Kc}.

A natural question is now whether we can do the same for a multiple coloring. Take a k-fold coloring of a graph Γ with n colors. Now each vertex is colored by exactly k out of n colors, so consider the graph we will denote K(n, k), whose vertices are all the k-subsets out of the n-set of used colors. We know that two adjacent vertices of Γ cannot share a color, so if we make all pairs of vertices of K(n, k) that are disjoint as subsets adjacent, there is a natural homomorphism from Γ into K(n, k).

Again, the converse is also clear, so we have the following equivalence:

Proposition 1.27. A k-fold coloring of a graph Γ with n colors is equivalent to a homomorphism from Γ into the graph K(n, k).

As a consequence:

Proposition 1.28. For each positive integer k we have:

χk(Γ) = min{n such that Γ → K(n, k)}.

This graph K(n, k) is called the Kneser graph, and we will say more about it in Chapter 3.

1.3 Finite projective spaces

1.3.1 Finite ﬁelds Deﬁnition 1.29 (Group). A group (G, ) is a set G with a binary operation such that

(i) for all a, b ∈ G: a b is an element in G,

(ii) for all a, b, c ∈ G: (a b) c = a (b c), 16 CHAPTER 1. INTRODUCTION

(iii) there is an element e ∈ G such that a e = e a = a for all a ∈ G (this element is called the identity element), and (iv) for each a ∈ G, there is an inverse element a0 such that aa0 = a0a = e. A group G for which a b = b a for all a, b ∈ G is called a commutative group. The number of elements of the group is called the order of the group.

Definition 1.30 (Field). A field (F, +, ·) is a set F together with two binary operations + and · such that (i) (F, +) is a commutative group with identity element 0 (called the additive group), (ii) (F∗, ·) is a commutative group with identity element 1 (called the multiplicative group), and (iii) for all a, b, c ∈ F: a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c where F∗ = F \{0}. The number of elements of the field is called the order of the field.

Remark 1.31. A set F with binary operations + and · that has the same properties as above, except that the group (F∗, ·) is not commutative, is called a division ring. Most of the times we will write just F instead of (F, +, ·).

Some well-known fields are the field of rationals Q, the field of real numbers R and the field of complex numbers C. Note that they all have infinite order.

A field with finite order is called a finite field. The following theorem is a well-known fact: Theorem 1.32. If F is a finite field with order q, then q = ph where p is a prime number and h ≥ 1. Moreover this field is unique (up to isomorphism) with this order.

We will denote the ﬁnite ﬁeld of order q by GF(q). 1.3. FINITE PROJECTIVE SPACES 17

Remark 1.33. If p is a prime then GF(p) is actually the set of integers modulo p, with the operations deﬁned by performing the operations in Z and taking the result modulo p. This ﬁeld is also denoted by Z/pZ or Fp.

Remark 1.34. The (little) theorem of Wedderburn states that all ﬁnite division rings are in fact ﬁelds.

Deﬁnition 1.35. The characteristic of a ﬁeld F is the smallest positive integer k, if it exists, such that a + a + ... + a = 0 (k repeated terms) for all a ∈ F and 0 otherwise. It is denoted by char(F ).

It is known that char(Q) = char(R) = char(C) = 0 and char(GF(q)) = p where q = ph with p a prime.

Definition 1.36. A subfield (F0, +, ·) of a field (F, +, ·) is a subset F0 of F that, together with the additive and multiplicative operations of F again forms a field.

For example: the field of rationals is a subfield of the field of reals, which is again a subfield of the complex numbers. For the finite fields we have the following:

h Proposition 1.37. GF(q1) is a subﬁeld of GF(q2) if and only if q1 = p 1 h and q2 = p 2 , for some prime p and h1|h2.

1.3.2 Projective spaces We will introduce projective spaces using their axiomatic description. After that, we will introduce the models of projective spaces that we will work with for the rest of this thesis.

First we need to know what a point-line incidence structure is:

Deﬁnition 1.38. A point-line incidence structure (sometimes also called a point-line geometry)(P, B, I) consists of two disjoint sets P and B. The elements of P are called points, and P is called the point set. The elements of B are called lines, and B is called the line set. I is a symmetric relation I ⊂ (P × B) ∪ (B × P), called the incidence relation. 18 CHAPTER 1. INTRODUCTION

If (p, L) ∈ I for a point p and a line L we can say that p is incident with L (and because I is symmetric, also the other way around). Most of the time we will use the more intuitive expressions “the point p lies on the line L” and “the line L goes through the point p”. Similarly, we will say that two lines intersect in a point if they are both incident with that point and that a line connects two points if those points are both incident with the line.

With some axioms, we can deﬁne a projective space:

Deﬁnition 1.39. A projective space is an incidence structure S = (P, B, I) that satisﬁes the following axioms:

(i) Any two distinct points p and q are connected by exactly one line (and we can denote this line by pq),

(ii) for any four distinct points a, b, c and d such that the line ab intersects cd we have that the line ac intersects bd, and

(iii) any line is incident with at least three points.

Now consider a line L of a projective space S = (P, B, I) and the set P0, the set of all points of P incident with L. Deﬁne the incidence structure L = (P0, {L}, I0), where I0 is the restriction of I to (P0 × {L}) ∪ ({L} × P0). It is clear that L is also a projective space, called a projective line. A projective space that is no projective line is called a non-degenerate projective space.

A projective plane is a projective space S = (P, B, I) that also satisﬁes a stronger version of axiom (ii) and a fourth axiom:

(i) Any two distinct points p and q are connected by exactly one line (and we can denote this line by pq),

(ii∗) any two lines intersect in at least one point,

(iii) any line is incident with at least three points, and

(iv) there are at least two lines.

Note that axiom (ii*) also holds for projective lines, that is why we need the extra axiom (iv).

Remark 1.40. There is an equivalent axiom system for projective planes: 1.3. FINITE PROJECTIVE SPACES 19

(i) Any two distinct points are connected by exactly one line,

(ii0) any two distinct lines intersect in exactly one point, and

(iii0) there exist four distinct points such that no three of those points are incident with the same line.

Now take a projective space S = (P, B, I). Because two points determine exactly one line, we can identify a line with the set of points incident with the line. Hence we can think of lines as subsets of P.

Deﬁnition 1.41. A set A ⊂ P is called linear if every line meeting A in at least two points is completely contained in A. A linear set containing at least two points must therefore contain at least one line. Denote the set of lines contained in A by B0, then it is not hard to see that S(A) := (A, B0, I0) is a non-degenerate projective space or a line (here I0 is the incidence relation I restricted to the set A). The incidence structure S(A) is called a (linear) subspace of S.

Like we identiﬁed lines with their point sets we can also identify every subspace by its point set and refer to both of them as subspace. It is obvious that the empty set, a singleton of P, a line and the set P itself are examples of subspaces. As every subset of P is clearly contained in at least one subspace, we can deﬁne the span of an arbitrary subset of P:

Deﬁnition 1.42. The span of a set B ⊆ P, denoted by hBi is deﬁned as:

\ hBi = {C|B ⊆ C,C is a linear set}.

It is clear that hBi is always a subspace.

Note that for two sets A, B ∈ P we will use hA, Bi as a notation for hA ∪ Bi.

A set of points A ⊂ P is called linearly independent if for any subset A0 ⊂ A and point p ∈ A \ A0, we have p 6∈ hA0i. In other words: hA0i T A = A0. 20 CHAPTER 1. INTRODUCTION

Deﬁnition 1.43. A basis of a projective space S is a linearly independent set of points that spans the entire space. It is not hard to show that every basis has the same number of points. This number is called the rank of S and will be denoted by rk(S).

One can deﬁne the rank of a subspace in exactly the same way. Take a point p. It is obvious that h{p}i = {p}, therefore {p} is a basis for the subspace with {p} as point set, hence a point is a rank-1 subspace. A line is spanned by two distinct points and the set consisting of those two points is linearly independent, so the rank of a line is 2. A projective plane can be shown to have rank 3 and every rank-3 subspace of a projective space can be shown to be a projective plane. The empty set will be considered as a subspace of rank 0.

We will deﬁne the projective dimension of a projective (sub)space S, to be the rank of S minus one3. We denote this by dim(S). An m-dimensional subspace (that is, a space of rank m + 1) will also be called an m-space. In that way, the empty set is a (−1)-space, a point is a 0-space, a line a 1-space and plane is a 2-space. If the projective space has dimension n, an (n − 1)-space in that space will be called a hyperplane.

The following formula is very useful to determine the dimension of the span or the intersection of two subspaces:

Theorem 1.44 (Dimension formula). If U and V are two subspaces of a projective space S, then

dim(hU, V i) + dim(U ∩ V ) = dim(U) + dim(V ).

Now we will state two theorems that are important in characterizing projective spaces. Consider a projective space S = (P, B, I) of dimension at least 2. Take any distinct points p1, p2, p3 and r1, r2, r3 of S for which the lines p1r1, p2r2 and p3r3 are concurrent in a point s, and such that no line pipj or rirj (for i, j ∈ {1, 2, 3} and i 6= j) contains s. Deﬁne the points

3The reason for that we want to conserve the intuition that a point has dimension 0, a line dimension 1, a plane dimension 2, etc. 1.3. FINITE PROJECTIVE SPACES 21 tij := pipj ∩ rirj for i, j = 1, 2, 3, i < j. Now the projective space S is called Desarguesian if and only if t12, t13 and t23 are collinear for all possible choices of pi and ri. This conﬁguration in a Desarguesian projective space is called a Desargues conﬁguration. See Figure 1.1.

Figure 1.1: Desargues conﬁguration

This criterion classiﬁes all projective spaces of dimension at least 3: Theorem 1.45. An n-dimensional projective space with n ≥ 3 is Desargue- sian. A proof of this theorem is given in [2].

Now take two distinct lines L and M and points li ∈ L and mi ∈ M for i = 1, 2, 3 all different from L ∩ M. Define the points tij := limj ∩ ljmi for i, j = 1, 2, 3, i < j. The projective space S is called Pappian if and only if t12, t13 and t23 are collinear for all possible choices of li and mi. The resulting configuration in a Pappian projective space is called a Pappus configuration. See Figure 1.2. 22 CHAPTER 1. INTRODUCTION

Figure 1.2: Pappus conﬁguration

A connection between Desarguesian and Pappian projective spaces is given in the following theorem which is proved in [26]:

Theorem 1.46. All Pappian projective spaces are also Desarguesian.

We will end the axiomatic description here and introduce the model we will work with for the rest of this thesis.

1.3.3 The projective space PG(n, q) Consider the (n + 1)-dimensional vector space V(n + 1,K) over the finite field K. More generally one can take a left vector space over a division ring. Define P as the set of 1-dimensional subspaces of V(n + 1,K) and B as the set of 2-dimensional subspaces of V(n + 1,K). Define I to be the symmetrised set theoretic inclusion. Now it is easy to check that (P, B, I) is a projective space of dimension n. This projective space will be denoted 1.3. FINITE PROJECTIVE SPACES 23 by PG(n, K). It is clear that an (r +1)-dimensional subspace of V(n+1,K) becomes an r-dimensional subspace of PG(n, q). Thus the rank of a projective subspace of PG(n, K) is equal to the dimension of the corresponding subspace of the underlying vector space, whereas the (geometric) dimension of a projective subspace is one less than the (vectorial) dimension of the corresponding subspace of the underlying vector space. For that reason we will also refer to the (vectorial) dimension of a subspace of the underlying vector space as the rank of that subspace.

One can deﬁne PG(n, K) in a more practical but equivalent way. Two vectors x, y of V(n + 1,K) \{0} are called equivalent if and only x = ky for some k ∈ K \{0}. Now the point set of PG(n, K) is just the set of all equivalence classes under this equivalence relation. The point that is the equivalence class of a vector x will be denoted by P (x), and x is called a coordinate vector of the point P (x). Points P (x1),...,P (xr) are linearly independent if the corresponding set of vectors is linearly independent in V(n + 1,K). A subspace of PG(n, K) of dimension r is a set of all points whose corresponding vectors form a (r + 1)-dimensional subspace of V(n + 1, q).

The reason that PG(n, K) is highlighted here is given by the following theorem: Theorem 1.47. Let S be a projective space. Then (i) S = PG(n, K) for some division ring K if and only if S is Desargue- sian. (ii) S = PG(n, K) for some ﬁeld K if and only if S is Pappian. From Theorem 1.45 it now follows that: Theorem 1.48. If S = (P, B, I) is a projective space of dimension at least 3, then S = PG(n, K) for some division ring K. Note that there exist a lot of projective planes (projective spaces of dimension two) that are not isomorphic to PG(n, K) for some division ring K.

If K is finite, it must be a field, according to the (little) Wedderburn theorem, hence K = GF(q) for some prime power q. In this case, the projective space PG(n, K) is also denoted by PG(n, q). But now Theorem 1.47 states that for finite projective spaces being Pappian is equivalent with them being Desarguesian. From this fact and Theorem 1.48 one can conclude the following: 24 CHAPTER 1. INTRODUCTION

Theorem 1.49. If S = (P, B, I) is a ﬁnite projective space of dimension at least 3, then S = PG(n, q) for some prime power q.

For the rest of this thesis we will only use finite projective spaces (and hence, over a finite field).

1.3.4 Some counting in PG(n, q) In this section we will determine some numbers that will play a role in the rest of this thesis.

A ﬁrst useful question is determining the number of k-dimensional subspaces in PG(n, q) (where −1 ≤ k ≤ n). This is the number of rank-(k + 1) subspaces of a rank-(n + 1) vector space.

So let us count the number of rank-k subspaces of V(n, q). We can use the same method that we used in Section 1.1 to calculate the number of k-subsets in an n-set. A rank-k subspace is spanned by k linearly independent vectors. For the first vector we have qn − 1 choices (the all-zero vector cannot be chosen). The second vector cannot lie in the rank 1-subspace defined by the first, so we have qn − q choices. For the third vector we have qn − q2. And so on. Finally for the k-th vector we have qn − qk−1. But the k chosen vectors are not unique to span this rank-k subspace. So we have to divide by the number of choices of vectors that span the same subspace. Using the same argument this is (qk −1)(qk −q) ··· (qk −qk−1). We thus have that the number of rank-k subspaces in a rank-n vector space over GF(q) is:

(qn − 1)(qn − q) ··· (qn − qk−1) hni = (qk − 1)(qk − q) ··· (qk − qk−1) k q

Definition 1.50 (Gaussian coefficient). The number of rank k-subspaces n in a rank n-vector space over GF(q) is given by the Gaussian coefficient k q.

We can restate this in terms of (projective) dimensions: The number of k-subspaces in an n-dimensional projective space over h i GF(q) is given by n+1 . k+1 q 1.4. FINITE POLAR SPACES 25

Another useful piece of combinatorial information is the number of k- dimensional subspaces containing a fixed l-dimensional subspace in PG(n, q), for −1 ≤ l < k < n. This is the same as the number of rank-(k+1) subspaces containing a fixed rank-(l + 1) subspace in V(n + 1, q). Now this is the same as the number of rank-(k − l) subspaces in the quotient V(n + 1, q)/V(l + 1, q) which is isomorphic to V(n − l, q). So we have that the number of k- dimensional subspaces containing a fixed l-dimensional subspace in PG(n, q) is the same as the number of (k − l − 1)-dimensional subspaces in PG(n − h i l − 1, q), which is n−l . k−l q

1.4 Finite polar spaces

In the previous section we encountered a first kind of incidence structure, namely the projective spaces (and planes). In this section we will define another type: the polar spaces. Definition 1.51. A (non degenerate) polar space of rank n (n ≥ 2) consists of a point set P together with a family of subsets of P, called the singular subspaces, satisfying the following axioms: (i) A singular subspace, together with the singular subspaces it contains is a k-dimensional projective space, for some −1 ≤ k ≤ n − 1. This dimension is by definition also the dimension of the singular subspace of the polar space.

(ii) The intersection of two singular subspaces is again a singular subspace.

(iii) Take a singular subspace U of dimension n − 1 and a point p ∈ P \ U. There is exactly one singular subspace V such that p ∈ V and U ∩ V has dimension n − 2. The singular subspace V contains all the points of U that lie on a common line with p.

(iv) There are at least two disjoint singular subspaces of dimension n − 1. Note that an incidence structure that satisfies axioms (i),(ii) and (iii) but not (iv) is called a degenerate polar space. The singular subspaces of (maximal) dimension n − 1 are called the generators of the polar space. A finite polar space is a polar space with a finite point set. As with the projective spaces, all polar spaces of rank at least 3 have been classified. In the (thick) finite case all polar spaces of rank at least 3 26 CHAPTER 1. INTRODUCTION are classical polar spaces. Here is a list of all the finite classical polar spaces with their ranks:

• Q+(2n+1, q) non-singular hyperbolic quadric in PG(2n+1, q) for some n ≥ 1, giving a hyperbolic polar space of rank n + 1.

• Q(2n, q): non-singular parabolic quadric in PG(2n, q) for some n ≥ 2, giving a parabolic polar space of rank n.

• Q−(2n + 1, q): non-singular elliptic quadric in PG(2n + 1, q) for some n ≥ 2, giving a hyperbolic polar space of rank n.

• W (2n + 1, q): polar space consisting the points of PG(2n + 1, q) together with the totally isotropic subspaces of a non-singular symplectic polarity of PG(2n + 1, q), giving a symplectic polar space of rank n.

• H(2n, q2) : non-singular Hermitian variety in PG(2n, q2) for some n ≥ 2, giving a Hermitian polar space of rank n.

• H(2n + 1, q2): non-singular Hermitian variety in PG(2n + 1, q2) for some n ≥ 1, giving a Hermitian polar space of rank n + 1.

The following theorem (see e.g. [28]) gives the size (number of points) of those ﬁnite classical spaces:

Theorem 1.52. The numbers of points of the ﬁnite classical polar spaces are given by:

|Q+(2n + 1, q)| = (qn + 1)(qn+1 − 1)/(q − 1), |Q(2n, q)| = (q2n − 1)/(q − 1), |Q−(2n + 1, q)| = (qn − 1)(qn+1 + 1)/(q − 1), |W (2n + 1, q)| = (q2n+1 − 1)/(q − 1), |H(2n, q2)| = (q2n+1 + 1)(q2n − 1)/(q2 − 1), |H(2n + 1, q2)| = (q2n+2 − 1)(q2n+1 + 1)/(q2 − 1).

In the case of rank 2, there are a lot of non-classical polar spaces known. Chapter 2

Combinatorics and q-analogues

A natural way of generalizing problems from extremal combinatorics to finite geometry is the q-analogue. This is done by changing the definition of the problem as follows: replace all occurrences of the words “set of size n” with “vector space over the field GF(q) with or rank n”. Of course some other words in the definition should also be changed accordingly, for example if the original problem talks about disjoint subsets, the q-analogous problem talks about subspaces that intersect trivially. In this chapter we give some examples of such generalizations by taking q-analogues of some classical problems in extremal combinatorics.

2.1 An easy example

In Section 1.1 we counted the number of k-subsets in an n-set. The q-analogue of this problem is counting the number of rank-k subspaces in V(n, q) and that is what we did in Section 1.3.4. Let us take a closer look at both results.

n The number of projective points in PG(n − 1, q) is given by 1 q. If we n+1 now recursively deﬁne [n + 1]q! = 1 q [n]q! and [0]q! = 1 we can write

hni [n] ! = q . k q [k]q![n − k]q!

27 28 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

Note that this notation looks a lot like the definition of the binomial coefficients. In that light it is not so strange that Gaussian coefficients are also called q-binomial coefficients.

An even stronger validation of this name is a phenomenon we see with a lot of q-analogues. Because we work with GF(q), the q’s used are prime powers. But if we look at the formulas as functions on R in the variable q, we can take the limit of these functions for q → 1+. For example the limit (by using de l’Hˆopital’s limit rule) of the number of rank 1-subspaces in a rank-n vector space is:

hni qn − 1 lim = lim q→1+ 1 q q→1+ q − 1 = n which is of course the number of elements in an n-set. Using this it is straightforward that

hni n lim = . q→1+ k q k

We will see that, in a lot of cases, taking the limit for q → 1+ of a value in the q-analogue of a problem gives us the value from the original problem. Note that this is not always the case; see, for example, Remark 3.9 in the next chapter.

2.2 Sperner’s Theorem

An important subdomain of extremal combinatorics is extremal set theory. The goal in this domain is to ﬁnd the maximum size of a family of sets satisfying certain assumptions. In this section, we consider Sperner families.

2.2.1 Original problem A classical theorem in extremal set theory is Sperner’s theorem. If we ﬁx a universe X with size n, the largest uniform family, that is a family in which n all sets have the same size, is the family of all k-subsets where k = 2 (or 2.2. SPERNER’S THEOREM 29

n n k = ). This uniform family has size n . In 1928 Sperner proved 2 b 2 c ([44]) that this is the best possible upper bound for the size of an antichain in general (for both uniform and non-uniform Sperner families) and that equality occurs exactly in the case just described.

There are several proofs known of this upper bound. Here we give one that uses the stronger LYM inequality. This inequality is named after D. Lubell [37], K. Yamamoto [48] and L.D. Meshalkin [39]. Here Lubell’s Per- mutation Method [37] is used to prove the inequality:

Lemma 2.1 (LYM inequality). If F is a Sperner family of subsets of a set X with |X| = n, then X 1 ≤ 1. n A∈F |A|

Proof. Consider the subset poset of X: (2X , ⊆). It is easy to see that the number of maximal chains in this poset is n! and that the number of maximal chains containing a certain subset A is |A|!(n − |A|)!.

Now count the pairs (A, C) where A ∈ F and C is a maximal chain that contains A. If we take a subset A in F, we have |A|!(n−|A|)! maximal chains that contain A. On the other hand, a maximal chain C can contain at most X one element of F (because F is an antichain). Thus |A|!(n − |A|)! ≤ n!. A∈F Dividing by n! gives the required result.

For other applications of the LYM inequality, see eg. [33].

Now we can prove Sperner’s Theorem:

Theorem 2.2. If F is a Sperner family of subsets of an n-set X, then

n |F| ≤ n . 2

Proof. Because n n n ≥ for all A ∈ F 2 |A| 30 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES we have, using the LYM inequality:

X 1 |F| 1 ≥ ≥ n n A∈F n |A| b 2 c

2.2.2 q-analogue Sperner’s theorem gives the maximum size of an antichain in the subset poset of a ﬁnite set. The natural q-analogue question therefore is: what is the maximum size of an antichain in the subspace poset of a ﬁnite vector space V(n, q)?

It turns out that the answer is what one would expect, and the proof is a complete q-analogue of the set case. We now state and prove the q-analogue of the LYM-inequality.

Lemma 2.3. If F is an antichain in the subset poset of V(n, q), then

X 1 ≤ 1. h n i A∈F rk(A) q

Proof. The number of maximal chains in this poset is [n]q! and the number of chains containing a subspace A is [rk(A)]q![n − rk(A)]q!. Counting the pairs (A, C) where A ∈ F and C is an antichain containing A yields: X [rk(A)]q![n − rk(A)]q! ≤ [n]q!. A∈F

Dividing both sides of this inequality by [n]q! gives the stated result.

n h n i If we now use this lemma together with the fact that k ≤ n for q b 2 c q all 0 ≤ k ≤ n we ﬁnd the q-analogue sought for.

Theorem 2.4. If F is an antichain in the subset poset of V(n, q), then

n |F| ≤ n . 2 q In this case equality holds if and only if F is the antichain consisting of n n all subspaces of rank 2 (or 2 ). 2.3. ERDOS-KO-RADO˝ THEOREM 31

2.3 Erd˝os-Ko-Radotheorem

Another classic result in extremal set theory is the Erd˝os-Ko-RadoTheorem.

2.3.1 Original problem Theorem 2.5 (Erd˝os-Ko-RadoTheorem (1961) [18]). If F is a k-uniform n family with a universe X of size n, where k ≤ 2 , and every pair of members of F intersect, then n − 1 |F| ≤ . k − 1

A family of mutually intersecting k-subsets of an n-set is called an (n, k)- EKR family. An EKR family that cannot be extended to another EKR family by adding subsets is called a maximal EKR family (or maximal intersecting family).

We do not give the original proof here, but we sketch a much more elegant proof by G. Katona in 1972 [31], which is inspired by Lubell’s Permutation Method.

Proof. Consider a labeling of an n-cycle by the elements of X. A path of length k in this labeled cycle corresponds to a k-subset of X. A collection of paths that pairwise overlap1 has size at most k, and in total there are n paths of length k on this cycle. So at most a fraction k/n of the k-subsets “occurring” in this cycle as paths mutually intersect. Each subset occurs the same number of times as a path among all possible labelings of an n-cycle n by elements of X. So at most a fraction k/n of the k k-subsets mutually intersect.

This bound is tight. We give some examples of families attaining this n−1 bound. There is an obvious example of a family of size k−1 , namely a family that consists of all k-subsets containing a common element of X. Such a family is called a point pencil and the common element is called the center of the point pencil. In Section 2.5 we will show that this is the only type of family attaining this upper bound for n > 2k.

1Here to overlap means to have at least a vertex in common. 32 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

If n = 2k, there are other examples besides point pencils. Each k-subset has exactly one complementary k-subset. An intersecting family can contain at most one k-subset out of each complementary pair. A family consisting 1 2k of exactly one k-subset out of each complementary pair has size 2 k which is the maximum size.

2.3.2 q-Analogue In the original Erd˝os-Ko-Radotheorem, F is a family of k-subsets of an n-set (with 2k ≤ n) such that no two members are disjoint. The statement of the theorem can be viewed in two ways:

n−1 1. k−1 is an upper bound on the size of F, or

|F| k 2. ≤ where Fk is the family of all k-subsets of this n-set (this is |Fk| n the statement that was proved by Katona).

In a q-analogue version of the Erd˝os-Ko-Radotheorem, F will be a family of rank-k subspaces of V(n, q) with 2k ≤ n in which no two members intersect trivially. We will call such a family an [n, k]q-EKR family. So now we can expect the following two statements:

h n−1 i (1q) |F| ≤ , and k−1 q

|F| k (2q) ≤ where Fk is the family of all rank k subspaces of V(n, q). |Fk| n

Because the number of rank-k subspaces containing the same rank-1 h n−1 i subspace is , the inequality in (1q) is best possible. Furthermore, we k−1 q have that h n−1 i k−1 qk − 1 k q = . n qn − 1 n k q

Therefore (1q) imposes a much stronger bound than (2q).

W.H. Hsieh [29] generalized Katona’s permutation method in 1973 to prove statement (2q), and, using a lot of counting arguments, he proved the stronger statement (1q) for n ≥ 2k + 1. 2.4. BOLLOBAS’S´ THEOREM 33

The case n = 2k remained open until 1986 when P. Frankl and R.M. Wilson [21] also proved the statement in this case.

Examples of families reaching this upper bound are easily found. If n > 2k, a family consisting of all rank-k spaces that contain the same rank- 1 subspace has this size. In the case n = 2k we have the same type of families and also the families that consist of all the rank-k spaces in a given hyperplane (rank-(2k − 1) subspace). In [21], Frankl and Wilson state they have a proof that those are the only families attaining the upper bound. In a paper, [24], of M.W. Newman and C. Godsil a short proof of this statement is given.

Remark 2.6. In [29] and [21], Hsieh, Frankl and Wilson actually prove the following, more general result.

Theorem 2.7. If F is a [n, k]q-EKR family such that the rank of the intersection of each pair of members is at least t (with n ≥ 2k − t), then ( ) n − t 2k − t |F| ≤ max , . k − t q k q

This result is used to improve some bounds on a similar generalization of the original Erd˝os-Ko-Radoproblem in [18].

2.4 Bollob´as’stheorem

For all of the problems we encountered in the previous sections, the answer in the original version was the limit case for q → 1+ of the answer in the vector space version. We already noted that this is not always true. For the following problem we see an even stronger connection between the diﬀerent versions.

2.4.1 Set version Helly’s Theorem gives a characterization of the dimension of a linear space over R in terms of intersection properties of convex sets: 34 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

n Theorem 2.8. If C1,C2,...,Cm ⊆ R are convex sets such that any n + 1 intersect, then all of them intersect.

One can wonder if other objects than convex sets obey similar laws. For example, it is easy to see that, if for each set of at most three edges of a graph we have that those edges have nonempty intersection, then all edges of the graph have nonempty intersection. This statement even generalizes over r-uniform set systems (note that a graph is a 2-uniform set system):

Proposition 2.9. If each family of at most r + 1 members of an r-uniform set system intersects, then all members intersect.

The statement about graphs can also be generalized in a diﬀerent direc- tion:

s+2 Theorem 2.10 ([17]). If each family of at most 2 edges of a graph can be covered by s vertices, then all edges can.

This was proved by P. Erd˝os,H. Hajnal and J. W. Moon in 1964. We will not give this proof but prove a more general result later in this section. The complete graph on s + 2 vertices shows that this result is sharp.

In 1965 B. Bollob´asproved ([4]) the following generalization of those problems:

r+s Theorem 2.11. If each family of at most s members of an r-uniform set system can be covered by s points, then all members can.

We will not prove this theorem, but a slightly more powerful theorem, also by Bollob´asin [4]:

Theorem 2.12. Let A1,...,Am and B1,...,Bm be subsets of size s of a set of size r such that

(i) Ai and Bi are disjoint for i = 1, . . . m, and

(ii) Ai and Bj intersect if i 6= j (1 ≤ i, j ≤ m).

r+s Then m ≤ r . There exist a lot of diﬀerent proofs of this theorem, for example by Jaeger-Payan ([30]) and Katona ([32]). We will give a proof by L. Lov´aszin [35]. First we need a proposition called the diagonal criterion: 2.4. BOLLOBAS’S´ THEOREM 35

Proposition 2.13. For i = 1, . . . , m, let fi : X → F, (where X is a set and F a ﬁeld) be functions and ai ∈ X elements such that

6= 0 if i = j; f (a ) i j = 0 if i 6= j.

X Then f1, . . . fm are linearly independent in the space F .

Pm Proof. Suppose that i=1 λifi = 0 is a linear relation on the fi’s. Substi- tuting aj for some 1 ≤ j ≤ m in the functions gives λjfj(aj) = 0, which implies that λj = 0. This is true for all j = 1, . . . , m, hence the functions are linearly independent.

Now we can give Lov´asz’sproof of Theorem 2.12:

Sm Proof. Deﬁne V = i=1(Ai ∪ Bi) and associate vectors

r+1 p(v) = (p0(v), p1(v), . . . , pr(v)) ∈ R to each v ∈ V such that any r + 1 of those vectors are linearly independent (note that there are constructive ways of doing this). Now we associate a polynomial fW (x) in r + 1 variables x = (x0, . . . , xr) to every W ⊆ V : Y fW (x) := (p0(v)x0 + p1(v)x1 + ... + pr(v)xr). v∈W

This is a homogeneous polynomial of degree |W | with:

6= 0 if x is orthogonal to none of the p(v), v ∈ W ; f (x) W = 0 otherwise.

Now take a vector aj orthogonal to the subspace spanned by the vectors corresponding to the elements of Aj for each j = 1, . . . , m. This guarantees that aj is orthogonal to p(v) if and only if v ∈ Aj. We can see that fBi (aj) = 0 if and only if Aj and Bi intersect, hence if and only if i 6= j. By the diagonal criterion this means that the polynomials fB1 , . . . , fBm are linearly independent in the space of homogeneous polynomials of degree s in r + 1 (r+1)+s−1 variables. Since this space has dimension s , we have that m ≤ r+s s . 36 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

2.4.2 Set vs. subspace version In the same paper where he proves Theorem 2.12, [35], Lov´aszalso proves the following result using a similar technique:

Theorem 2.14. Let U1,...,Um be r-dimensional subspaces and B1,...,Bm be s-elements sets in a linear space W over the ﬁeld F such that

(i) Ui and Bi are disjoint for i = 1, . . . m, and

(ii) Ui and Bj intersect if i 6= j (1 ≤ i, j ≤ m).

r+s Then m ≤ r .

2.4.3 Subspace version Using tensor product methods Lov´aszgives another proof of theorem 2.12 in [35] and generalizes this proof to show the following result:

Theorem 2.15. Let U1,...,Um be r-dimensional subspaces and V1,...,Vm be s-dimensional subspaces of a linear space W over the ﬁeld F such that

(i) Ui ∩ Vi = 0 for i = 1, . . . m, and

(ii) Ui ∩ Vj 6= 0 if i 6= j (1 ≤ i, j ≤ m).

r+s Then m ≤ r

We see that in all three versions (set-set, set-subspace and subspace- subspace) the answer of the problem is the same. Our observation that in many cases the answer in the set version of a problem is a limit case of the answer in the vector space version is trivially true here.

2.5 The Hilton-Milner theorem

2.5.1 Original problem In Section 2.3, a tight upper bound on the size of an (n, k)-EKR family was shown. We saw that a point pencil is a family that attains this bound. In [27], A.J.W. Hilton and E.C. Milner give two examples of maximal (n, k)- EKR families that are not point pencils. 2.5. THE HILTON-MILNER THEOREM 37

Example 2.16. Take the set X = {1, 2, . . . n} and A = {1, 2, . . . k} with n 2 ≤ k ≤ 2 . The family F1 consisting of all k-subsets of X containing the element n and intersecting A, together with A, is clearly a maximal (n, k)- EKR family. Suppose that this is a point pencil, say with center x. Because x ∈ A we have that 1 ≤ x ≤ k, but the set {1, 2, . . . , k, n}\{x} is an element of F1 that does not contain x, a contradiction. Hence F1 cannot be a point n−1 n−k−1 pencil. The size of F1 is k−1 − k−1 + 1.

Example 2.17. The family F2, consisting of all k-subsets of X containing at least two elements of {1, 2, 3}, is a maximal intersecting family that is no point pencil. If k = 2, F2 has the same structure as F1. If k ≥ 3, it has size n−3 n−3 3 k−2 + k−3 , which is the same as |F1| if k = 3 and smaller than |F1| if k > 4.

In the same paper Hilton and Milner prove that F1 (and F2 if k = 2, 3) has the maximum size a maximal (n, k)-EKR family can have if it is not a point pencil:

n Theorem 2.18. If F is a (n, k)-EKR family, with 2 ≤ k ≤ 2 , that is no point pencil and not contained in one, then n − 1 n − k − 1 |F| ≤ − + 1. k − 1 k − 1 If n > 2k and this upper bound is attained, then F has the structure described in Example 2.16

Note that if n = 2k this is bound is the same bound as in the EKR theorem, because in that case, there were other maximal EKR families than point pencils that attain the EKR bound, too.

If n > 2k, this theorem is of course enough to conclude the following:

Corollary 2.19. If F is an (n, k)-EKR family, with n > 2k and

n − 1 n − k − 1 |F| > − + 1, k − 1 k − 1 then F is a point-pencil and hence n − 1 |F| = . k − 1 38 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

2.5.2 The q-analogue

We are looking for maximal [n, k]q-EKR families that are no point pencils (and no hyperplanes if n = 2k).

For k = 2, the only possibility is a family consisting of all rank-2 spaces (projective lines) in a rank-3 space (projective plane):

Proposition 2.20. Every [n, 2]q-EKR family, with n ≥ 4 F is contained in a point pencil or a projective plane

Proof. Suppose F is an [n, 2]q-EKR family that is not contained in a plane. Take two lines l1 and l2, they intersect, say in the point x and span a plane, say α. Take another line l3 of F, not in α. This line intersects l1 and l2 in respectively y1 and y2. Now y1 and y2 are the same point, otherwise l3 would lie in α. So the point y1 = y2 lies both on l1 and l2, hence it must be x. Now take another line l4 in F. Because l1, l2 and l3 are not coplanar (they do not lie on the same plane), at least one pair of them spans a plane that does not contain l4. Now repeat the same reasoning as above with those two lines and l4 to conclude that x is on l4. Continue this reasoning to conclude that all lines of F go through x, so F is contained in the point pencil with center x.

For k ≥ 3 the situation is much more complex. There are lots and lots of types of maximal EKR-families that are no point pencil (or hyperplane if n = 2k). In what follows we develop a description of the structure of the largest of those families (if n ≥ 2k + 1).

First, note that a point pencil has

n − 1 ≈ q(n−k)(k−1) k − 1 q elements.

Consider the following large examples.

Example 2.21. Let F3 be the family of all rank-k spaces (projective (k−1)- spaces) on a ﬁxed rank-2 space (projective line). This [n, k]q-EKR family has n − 2 ≈ q(n−k)(k−2) k − 2 q 2.5. THE HILTON-MILNER THEOREM 39 elements. Note that this family is not maximal, it can, for instance, be extended to a point pencil or to a family described in the next example.

Example 2.22. Let F4 be the family of all (k − 1)-spaces that intersect a given plane in at least a line. This family has size approximately n − 2 (q2 + q + 1) ≈ q(n−k)(k−2)+2. k − 2 q We can still ﬁnd larger examples

Example 2.23. Let F5 be the family of all (k − 1)-spaces that contain a ﬁxed point p and meet a ﬁxed (k − 1)-space π (that does not contain p), together with π. This family has size approximately k n − 2 ≈ q(n−k)(k−2)+k−1. 1 q k − 2 q Note that this is not a maximal family, but if we add all the (k − 1)-spaces in hp, πi that do not contain p, it is maximal. Doing that only adds ap- k proximately q elements to F5. So the order of magnitude of the size does not change. Note that this is the q-analogue of the Hilton-Milner example (Example 2.16). We will call this example the q-Hilton-Milner example.

Example 2.24. Take two (k − 1)-spaces π1 and π2 and a point p not contained in them such that π2 6⊂ hp, π1i. In that case both π1 \ π2 and π2 \ π1 k−1 contain approximately q points. Let F6 be the family of all (k−1)-spaces that contain p and meet both π1 \ π2 and π2 \ π1, together with π1 and π2. This family has approximate size n − 3 q2(k−1) ≈ q(n−k)(k−2)−(n−k)+2k−2. k − 3 q

Again, here we can add all (k − 1)-spaces that contain p and meet π1 ∩ π2, and they add at most approximately k − 1 n − 2 ≈ q(n−2)(k−2)+k−2 1 q k − 2 q to the previous number. Note that if n = 2k + 1, q(n−k)(k−2)−(n−k)+2k−2 = (n−k)(k−2)+k−3 q , so in that case, if π1 and π2 intersect in a (k − 2)-space, the second term dominates and we have an example that is of almost the same order of magnitude than Example 2.23. Note that also here we can add the (k − 1)-spaces that are contained in hp, π1i ∪ hp, π2i but do not contain p without changing the order of magnitude of the size of this family. 40 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

Remark 2.25. In both Examples 2.23 and 2.24 (the two largest we have found so far) there is a point that contains almost all (k − 1)-sets of the family.

The following lemma assures the existence of points with a high degree if there is a subspace that meets all elements of an EKR-family (the degree of a subspace is the number of elements of F it is contained in).

Lemma 2.26. Let F be an [n, k]q EKR-family. If there is an (l − 1)-space Λ that meets F (by which we mean that it meets all elements of F), then h i there is a point contained in at least |F|/ l elements of F and this point 1 q is contained in Λ.

Proof. Label the points of Λ as p1, p2, . . . , p l and deﬁne di as the number [ 1 ]q of elements of F that contain the point pi. Counting the pairs (p, π), where p is a point of Λ and π and element of F that contains p, in two ways yields

l [ 1 ]q X di ≥ |F|. i=1

Deﬁne d as the maximum of all di, then we have the required result for all points that attain this maximum.

We would like to prove a q-analogue of Theorem 2.18. We propose, for reasons that will become clear later, a lower bound for the families F we want to consider:

2 n − 2 q + q + 1 3k−n |F| ≥ 2 q . k − 2 q (q − 1)

For n = 2k + 1, this bound has a size of the same order of magnitude as the size of the q-Hilton-Milner example. For n > 2k + 1, the size is at least one order of magnitude smaller than the size of the q-Hilton-Milner example.

Depending on the value of q and n, this bound is larger or smaller than the size of the q-Hilton-Milner example. Let us consider some cases.

• Case 1: If q ≥ 4 and n ≥ 2k + 2; q = 3 and n ≥ 2k + 3 or q = 2 and n ≥ 2k + 4, the bound on F is smaller than the size of the q-Hilton- Milner example. 2.5. THE HILTON-MILNER THEOREM 41

• Case 2: If q ≥ 4 and n = 2k + 1; q = 3 and n = 2k + 2 or q = 2 and n = 2k + 3, the order of magnitudes are equal but the bound is larger than the size of the q-Hilton-Milner example.

• Case 3: If q = 3 and n = 2k + 1 or q = 2 and 2k + 1 ≤ n ≤ 2k + 2, the order of magnitue of the bound is larger than that of the size of the q-Hilton-Milner example. Let us start with the ﬁrst case.

Case 1

So deﬁne a type 1 family as a maximal [n, k]q-EKR family F, with k ≥ 3 and the given bounds on n and q, such that

2 n − 2 q + q + 1 3k−n |F| ≥ 2 q . k − 2 q (q − 1) The following lemma shows that there is always a line meeting such F. Lemma 2.27. If there is an (l − 1)-subspace meeting a type 1 family F and 3 ≤ l ≤ k, then there is an (l − 2)-subspace meeting F. h i Proof. Lemma 2.26 guarantees a point p with degree at least |F|/ l . Now 1 q either all elements of F contain this point or there is an F1 ∈ F that does not contain this point. In the former case F is a point pencil and all (l − 2)- spaces on the center of this point pencil meet F. In the latter case we can repeat the argument in the proof of Lemma 2.26 for p and F and find that there is a point q in F1, and hence a line L = pq, with degree at least h l i h k i |F|/ 1 1 . Now either all elements of F intersect L or there is an F2 ∈ F that is disjoint from L. In the former case all (l − 2)-spaces containing this line meet F. In the latter case we can find a plane π with degree at least h i h i2 |F|/ l k . Continuing this argument we either find at some point an 1 q 1 q (l − 2)-space meeting F, or we get an (l − 1)-subspace with degree at least h i h il−1 h i |F|/ l k . Since an (l −1)-space is contained in n−l (k −1)-spaces 1 q 1 q k−l q of PG(n, q), we have a contradiction if n − l |F| < . h i h il−1 k − l q l k 1 q 1 q 42 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

This is equivalent with

l−1 Y qn−i − 1 (q − 1)l(q2 + q + 1)q3k−n > 1. (qk−i − 1)(qk − 1) (ql − 1)(qk − 1)(q − 1)2 i=2 Now for 2 ≤ i ≤ l − 1 ≤ k − 1 we have

qn−2k(qk−i − 1)(qk − 1) = qn−i − qn−k − qn−k−i + qn−2k < qn−i − qn−k < qn−i − 1 so we get a contradiction if

(q − 1)l(q2 + q + 1)q3k−n q(n−2k)(l−2) > 1. (2.1) (ql − 1)(qk − 1)(q − 1)2 Since substituting l by l + 1 in the right hand side of this inequality is equivalent with multiplying this expression by

qn−2k(q − 1)(ql − 1) > 1, ql+1 − 1 we only need to get a contradiction for the smallest l, which is l = 3. Substitute for l = 3 in the right hand side of inequality 2.1 yields:

qn−2k(q − 1)3(q2 + q + 1)q3k−n qk = > 1. (q3 − 1)(qk − 1)(q − 1)2 qk − 1 So for l = 3, inequality 2.1 holds and we obtain a contradiction.

Since all elements of F satisfy the criterion of this lemma for l = k, there is indeed a line meeting F. Such a line that meets F will be called a hitting line.

With this knowledge we can improve our bound if n ≥ 3k. Corollary 2.28. If F is a type 1 family, then n − 2 |F| ≥ . k − 2 q Proof. Lemma 2.27 guarantees the existence of a hitting line of F. Since F is maximal, all (k − 1)-spaces through that hitting line must be elements of F. 2.5. THE HILTON-MILNER THEOREM 43

Note that for n ≥ 3k this bound is indeed larger than our assumed lower bound.

We prove some results about hitting lines.

Lemma 2.29. If F is a [n, k]q-EKR family, and L1 and L2 are hitting lines of F that intersect in a point p. Then all lines L through p in the plane hL1,L2i are also hitting lines.

Proof. Take such a line L. Since each element of F intersects both L1 and L2, it intersects the plane hL1,L2i in a line, and that line intersects L. Note that if L were not to lie on p then it would not intersect with the elements of F that intersect both L1 and L2 in p.

If there are three hitting lines in a triangle, we can characterize F.

Lemma 2.30. If L1,L2,L3 are non concurrent hitting lines of an [n, k]q- EKR family F in a plane α, then F is a family of (k − 1)-spaces that meet a ﬁxed plane (in this case α) in at least a line (see Example 2.22).

Proof. Each element of F has either 2 or 3 intersection points with the triple (L1,L2,L3). In the former case those two points determine the line in which the element intersects α and in the latter case α is contained in the element.

Deﬁne a type 1’ family F as a type 1 family that allows no plane such that all elements of F intersect it in at least a line. The following lemma gives a condition for a line to be a hitting line.

Lemma 2.31. Let F be an [n, k]q-EKR family. If a line L has a degree h i h i larger than k n−3 then it intersects F. 1 q k−3 q

Proof. Suppose that there is an element F ∈ F that does not intersect L. That means that there is a point p on F , and hence a plane hp, Li, that h i intersects more than n−3 elements of F, a contradiction. k−3 q

The following lemma states that if F is a type 1 family that is large enough, unless F is (part of) a point pencil, a hitting line always intersects at least one other hitting line. 44 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

Lemma 2.32. Let F be a type 1 family. If L is a hitting line and no line h i h i meeting L has degree more than k n−3 , then either 1 q k−3 q ! n − 2 k k n − 3 |F| ≤ + (q + 1) − 1 , k − 2 q 1 q 1 q k − 3 q or F is contained in a point pencil.

Proof. Let F ∈ F be such that F ∩ L = {p0}. If such an element does not exist then L is contained in all elements and F is contained in a point pencil. Now count the number of elements of F not containing L. For a point p 6= p , p ∈ L and any point q ∈ F , the line pq has degree at most h i h i 0 k n−3 . Since every element of F meets F we have that there are at 1 q k−3 q h i h i h i most k − 1 k n−3 elements of F through p, for all points p on 1 q 1 q k−3 q L diﬀerent from p0. Now if all elements of F contain p0, F is contained in a point pencil. So 0 assume that there is an element F of F that does not contain p0. The h i h i h i same argument yields that there are also at most k − 1 k n−3 1 q 1 q k−3 q h n−2 i elements of F through p0. Finally, there are at most elements of F k−2 q that contain L. Adding this up yields the desired result.

Remark 2.33. The bound in the previous lemma is approximately

q(n−k)(n−2) + q(n−k)(k−2)−n+3k−1.

• If n ≥ 3k, the ﬁrst term of this bound dominates. Because of Corollary 2.28 we cannot hope for a smaller bound, since we can simply take all (k − 1)-spaces through a hitting line. So at this point, we need ! n − 2 k k n − 3 |F| > + (q + 1) − 1 . k − 2 q 1 q 1 q k − 3 q

• If n = 3k − 1, this bound roughly doubles the previously assumed bound.

• If n < 3k − 1, the second term dominates and the new bound is a lot smaller than the previous bound. 2.5. THE HILTON-MILNER THEOREM 45

An intersection point of two hitting lines of a type 1’ family has large degree.

Lemma 2.34. Let F be a type 1’ family and let L1,L2 be two hitting lines 2 h k i h n−3 i with L1 ∩ L2 = {p}, then there are at most q elements of F 1 q k−3 q that do not contain p.

Proof. Suppose there is a line L in hL1,L2i that does not contain p and h i h i has degree more than k n−3 . By Lemma 2.31 L is a hitting line, but 1 q k−3 q we assumed that such a conﬁguration does not exist. That means that all such lines have degree at most this number. Now we count the number of elements of F that do not contain p. Each such element intersects the plane hL1,L2i in a line not through p. We just saw that such lines have a degree h i h i at most k n−3 . This yields the result. 1 q k−3 q A point with this property we will call a point with large degree. The previous lemmas give us a lower bound on F for the existence of such points. Corollary 2.35. If F is a type 1’ family such that ! n − 2 k k n − 3 |F| > + (q + 1) − 1 , k − 2 q 1 q 1 q k − 3 q then there is a point of large degree. Proof. Lemma 2.32, together with Lemma 2.27 guarantee the existence of two intersecting hitting lines. Now Lemma 2.34 guarantees that the intersection point of those two lines has a large degree.

The Lemma 2.34 has strong implications on the structure of hitting lines. Lemma 2.36. Two hitting lines of a type 1’ family F always meet.

Proof. Suppose that L1 and L2 are two disjoint hitting lines. If there is a hitting line meeting L1, by Lemma 2.34 the intersection point has degree at least 2 n − 2 q + q + 1 3k−n 2 k n − 3 (n−k)(k−2)+3k−n 2 q − q ≈ q . k − 2 q (q − 1) 1 q k − 3 q

If there is also a hitting line intersecting L2, this intersection point has also at least this degree. But then we have that k n − 3 n − 2 |F| ≥ 2|F | − 2q2 − , 1 q k − 3 q k − 2 q 46 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES or ! 1 k n − 3 n − 2 |F| ≤ 2q2 + . 3 1 q k − 3 q k − 2 q Since n ≥ 2k + 1, the second term dominates. If n < 3k, this bound is smaller than the lower bound we assumed for F, a contradiction. If n ≥ 3k, Corollary 2.28 gives a lower bound on F that is approximately 3 times higher than the upper bound we find here, which is also impossible. So we can assume that all lines meeting L are non hitting lines and hence h i h i 2 have degree less than k n−3 . So we can apply Lemma 2.32 and find 1 q k−3 q that ! n − 2 k k n − 3 |F| ≤ + (q + 1) − 1 . k − 2 q 1 q 1 q k − 3 q Actually, since all elements of F containing L must also intersect L , the 2 h i 1 first term of the left hand side can be reduced to (q + 1) n−3 . But now k−3 q the second term is always dominating and since this is always some orders of magnitudes smaller than the assumed lower bound on F we have a contradiction.

Let us now consider the next case.

Case 2

In this case the lower bound we assumed on F is larger than the size of the q-Hilton Milner example. That is why we will propose another bound that is slightly smaller than that example. Deﬁne a type 2 family F as a maximal [n, k]q-EKR family for the given bounds on n and q, such that

k n − 2 k n − 3 |F| ≥ − q . 1 q k − 2 q 2 q k − 3 q

In this case we can also prove the existence of a hitting line (or a hitting plane in the case q = 2) in the same way as before.

Lemma 2.37. Let F be a type 2 family. If there is an (l − 1)-subspace meeting F and 3 ≤ l ≤ k (or 4 ≤ l ≤ k if q = 2), then there is an (l − 2)- subspace meeting F. 2.5. THE HILTON-MILNER THEOREM 47

Proof. Using the same arguments as in Lemma 2.27 we ﬁnd a contradition if |F| l−1 > 1, h n−l i h l i h k i k−l q 1 q 1 q or if l−1 ! Y qn−i − 1 k (q − 1)l (qk−i − 1)(qk − 1) 1 (ql − 1)(qk − 1) i=2 q l−1 ! Y qn−i − 1 k (q − 1)l − q > 1. (qk−i − 1)(qk − 1) 2 (ql − 1)(qk − 1)2 i=3 q By using inequality 2.1 from Lemma 2.27 we can simplify this to obtain a contradiction if  h i  ! q k (qk−2 − 1) k (q − 1)l 2 q(n−2k)(l−2) 1 − q  > 1, l k  h k i  1 q (q − 1)(q − 1) (qn−2 − 1) 1 q or if   h k i k−2 l−1 1 − 1 (q − 1) (n−2k)(l−2) (q − 1)  q  q 1 −  > 1. ql − 1  (q + 1)(qn−2 − 1) 

Now since k qk − 1 − 1 < 1 q q − 1 the following inequality is needed for a contradiction: (q − 1)l−1 (qk − 1)(qk−2 − 1) q(n−2k)(l−2) 1 − > 1. (2.2) ql − 1 (q + 1)(q − 1)(qn−2 − 1) 1. If n = 2k + 1, and hence q ≥ 4, substituting l + 1 for l is equivalent with multiplying the right hand side of inequality 2.2 with q(q − 1)(ql − 1) > 1. ql+1 − 1 Therefore it is enough to obtain a contradiction for l = 3. In this case a contradiction is obtained if q(q − 1)2 (qk − 1)(qk−2 − 1) 1 − > 1. q3 − 1 (q + 1)(q − 1)(q2k−1 − 1) Carefully checking this shows that this inequality is true for q ≥ 4. 48 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

2. If n = 2k + 2, and hence q = 3, substituting l + 1 for l is equivalent with multiplying the right hand side of inequality 2.2 with q2(q − 1)(ql − 1) > 1. ql+1 − 1 Therefore it is enough to obtain a contradiction for l = 3. In this case a contradiction is obtained if q2(q − 1)2 (qk − 1)(qk−2 − 1) 1 − > 1. q3 − 1 (q + 1)(q − 1)(q2k−1 − 1) Carefully checking this shows that this inequality is true for q = 3.

3. If n = 2k + 3 and hence q = 2 substituting l + 1 for l is equivalent with multiplying the right hand side of inequality 2.2 with q3(q − 1)(ql − 1) > 1. ql+1 − 1 Therefore it is enough to obtain a contradiction for l = 3. In this case a contradiction is obtained if q3(q − 1)2 (qk − 1)(qk−2 − 1) 1 − > 1. q3 − 1 (q + 1)(q − 1)(q2k−1 − 1) Carefully checking this shows that this inequality is not true for q = 2. So we look at the case l = 4. In this case a contradiction is obtained if q6(q − 1)3 (qk − 1)(qk−2 − 1) 1 − > 1. q4 − 1 (q + 1)(q − 1)(q2k−1 − 1) Carefully checking this shows that this inequality is not true for q = 2.

In the same way as Corollary 2.28 one proves the following. Corollary 2.38. If F is a type 2 family with q > 2, then n − 2 |F| ≥ . k − 2 q

Now since Lemmas 2.29, 2.30 and 2.31 are stated for general [n, k]q- EKR families, they still hold (for q > 2). A quick check shows that the other results in Case 1 still hold in Case 2 for q > 2. Let us now continue with the last case. 2.5. THE HILTON-MILNER THEOREM 49

Case 3 Here we will use the same lower bound for F as in Case 2. Deﬁne a type 3 family F as a maximal [n, k]q-EKR family for the given bounds on n and q, such that

k n − 2 k n − 3 |F| ≥ − q . 1 q k − 2 q 2 q k − 3 q

In this case we can only prove the existence of a hitting plane for q = 3.

Lemma 2.39. Let F be a type 3 family with q = 3. If there is an (l − 1)- subspace meeting F and 4 ≤ l ≤ k, then there is an (l − 2)-subspace meeting F.

Proof. The proof is completely analoguous to the proof of Lemmas 2.27 and 2.37 with l = 4.

We summarize everything.

Main theorem Remark 2.33 shows that the cases n ≥ 3k − 1 and n < 3k − 1 should be treated a little bit diﬀerent.

The following lemma summarizes what we know about the existence of hitting lines in the diﬀerent cases.

Lemma 2.40. Hitting lines have the following properties:

1. Let F be a maximal [n, k]q-EKR family with n ≥ 3k − 1.

(a) If ! n − 2 k k n − 3 |F| > + (q + 1) − 1 k − 2 q 1 q 1 q k − 3 q

there exist at least two hitting lines. (b) If 2 n − 2 q + q + 1 3k−n 2 q ≤ |F| k − 2 q (q − 1) 50 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

and ! n − 2 k k n − 3 |F| ≤ + (q + 1) − 1 k − 2 q 1 q 1 q k − 3 q there exists at least one hitting line.

2. Let F be a maximal [n, k]q-EKR family with n = 2k + d, for 1 ≤ d < k − 1 (such that n < 3k − 1).

(a) If q ≥ 4 and n ≥ 2k + 2; q = 3 and n ≥ 2k + 3; or q = 2 and n ≥ 2k + 4, and

2 n − 2 q + q + 1 3k−n |F| ≥ 2 q k − 2 q (q − 1) there exist at least two hitting lines. (b) If q ≥ 4 and n = 2k + 1 or q = 3 and 2k + 2, and k n − 2 k n − 3 |F| ≥ − q 1 q k − 2 q 2 q k − 3 q there exist at least two hitting lines.

Now we investigate the structure of F if there are at least two hitting lines. Because of Lemma 2.36 we know that all hitting lines pairwise intersect. If there is no plane such that all elements of F intersect this plane in at least a line, then this means that all hitting lines intersect in the same point p, and this is a point of high degree. Now all the hitting lines span an m-dimensional subspace H for some m. Lemma 2.29 now guarantees that all lines in H that contain p are also hitting lines. So the existence of two hitting lines gives us the following structure: if there is no plane such that all elements of F intersect that plane in at least a line, there is an m-dimensional subspace H, with m ≥ 2, and a point p in H, such that all lines through p in H are exactly all the hitting lines of F. If all elements of F contain p, F is a point pencil. Otherwise m ≤ k and an element F of F that does not contain p hits H in at least an (m − 1)-dimensional subspace, since it must intersect all hitting lines. That means that two such elements intersect in at least an (m − 2)-subspace. Deﬁne F ∗ as the subset of F of elements that do not contain p. The previous argument shows that n − m |F ∗| ≤ qm . k − m q 2.5. THE HILTON-MILNER THEOREM 51

If m ≥ 3, this is a lot smaller than the size of F, if l = 2 Lemma 2.34 shows that F ∗ is a lot smaller than F. In both cases the size of F ∗ is negligible compared to the size of F. Now we have enough information to determine a new upper bound on the size of F: hmi n − 2 n − m |F| ≤ + qm if m ≥ 3, and 1 q k − 2 q k − m q hmi n − 2 k n − 3 |F| ≤ + q2 if m = 2. 1 q k − 2 q 1 q k − 3 q Note that if m = k, this structure is the structure of the q-Hilton-Milner example. If m < k, consider hF, pi for an element F ∈ F that does not contain p such that hF, pi contains H. It is clear that F meets H in an (m − 1)-subspace L. If now all elements of F are also contained in hF, pi, than m = k, so there is another element F 0 ∈ F not contained in hF, pi. Now F 0 also meets H in an (m − 1)-subspace L0 and F ∩ F 0 is at most an (m − 2)-space. If the intersection is an (m − 2)-space, the point p can have degree at most k − 1 n − 2 k 2 n − 3 + . 1 q k − 2 q 1 q k − 3 q If n < 3k −1, write n = 2k +d, for 1 ≤ d < k −1. Then we assumed that the order of magnitude of the size of F is at least q(n−k)(k−2)+k−d. We found that the size of F must have an order of magnitude at most q(n−k)(k−2)+m−1. Therefore k − d ≤ m − 1, or m ≥ k − d + 1. In particular, if n = 2k + 1, that means that m = k and we have the q-Hilton-Milner example.

We summarize everything in the following theorem.

Theorem 2.41. Let F is a maximal [n, k]q-EKR family and one of the following holds: 1. n ≥ 3k − 1 and ! n − 2 k k n − 3 |F| > + (q + 1) − 1 , k − 2 q 1 q 1 q k − 3 q

2. n = 2k + d, for 1 ≤ d < k − 1 (such that n < 3k − 1), 2 n − 2 q + q + 1 3k−n |F| ≥ 2 q , k − 2 q (q − 1) and one of the following hold: 52 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES

(a) q ≥ 4 and n ≥ 2k + 2, (b) q = 3 and n ≥ 2k + 3, or (c) q = 2 and n ≥ 2k + 4, 3. n = 2k + d, for 1 ≤ d < k − 1 (such that n < 3k − 1), k n − 2 k n − 3 |F| ≥ − q , 1 q k − 2 q 2 q k − 3 q and one of the following hold: (a) q ≥ 4 and n = 2k + 1, or (b) q = 3 and n = 2k + 2. Then F is one of the following: •F is a point pencil, and n − 1 |F| = . k − 1 q

• There is a plane such that all elements of F intersect that plane in at least a line, in that case F has approximate size n − 2 (q2 + q + 1) . k − 2 q

•F has the structure described above for some 2 ≤ m ≤ k, and hmi n − 2 n − m |F| ≤ + qm if m ≥ 3, and 1 q k − 2 q k − m q n − 2 k n − 3 |F| ≤ (q + 1) + q2 if m = 2. k − 2 q 1 q k − 3 q If m = k, F is a q-Hilton-Milner family. In the following corollary we state the situation for n = 2k + 1 explicitly since we need this in the next chapter.

Corollary 2.42. Let F be a maximal [2k + 1, k]q-EKR family, for q ≥ 4, such that k 2k − 1 k 2k − 2 |F| ≥ − q , 1 q k − 2 q 2 q k − 3 q Then F is one of the following: 2.6. SMALL MAXIMAL CLIQUES 53

•F is a point pencil, and

2k |F| = . k − 1 q

• There is a plane such that all elements of F intersect that plane in at least a line, in that case F has approximate size

2k − 1 (q2 + q + 1) . k − 2 q

•F is a q-Hilton-Milner family and

k 2k − 1 |F| ≤ + qk. 1 q k − 2 q

2.6 Small maximal cliques

In a previous section we studied the maximal size of maximal intersecting families of subsets of a set, and the q-analogous question of the maximal size of a maximal intersecting family of subspaces of a vector space. In this section we will study the other end of the spectrum. The main question here is what is the minimal size of a maximal intersecting family of ﬁnite sets (or subspaces of some ﬁnite vector space).

A k-clique is a collection of pairwise intersecting k-sets. A k-clique is maximal if it is not possible to add a new k-set that intersects all of the k-sets in the clique. The support of a clique is the union of all elements of the sets in the clique. A blocking set of a k-clique is a set that intersects all the k-sets of that clique.

It’s easy to see that if each blocking set of a k-clique either is an element of the clique or its size is larger than k, then the clique is maximal.

The minimal number of k-sets in a maximal k-clique will be denoted by m(k).

It is trivial to determine m(2). Take two intersecting 2-sets, say {0, 1} and {0, 2}; it is clear that this is not maximal. To add a third 2-set, we have 2 possibilities: we can add a new element 3 and take the pair {0, 3}, or take 54 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES the pair {1, 2}. It is clear that in the former case, we can add more new sets by adding new elements, while in the latter case, the clique is maximal. Therefore m(2) = 3.

The following case, m(3), is a bit more work, but can also be determined:

Proposition 2.43. m(3) = 7.

Proof. Consider the Fano plane PG(2, 2), this is a conﬁguration of 7 pairwise intersecting lines of size 3 and hence a 3-clique. Because the smallest blocking sets of a projective plane are lines, this clique is maximal. This gives m(3) ≤ 7. Explicit checking of all cases shows that smaller examples do not exist.

We saw that in this case the projective plane of order 2 gives an upper bound for m(3). In the same way a projective plane of order n gives a maximal n + 1-clique. This bound, and the other known bounds for m(k) are given in the following overview:

8 Theorem 2.44. 1. m(k) ≥ 3 k − 3 (Erd˝osand Lov´asz[19]), 2. m(k) ≤ k2 − k + 1 if a projective plane of order k − 1 exists (Meyer [41]),

3. m(2k) ≤ 3k2 (F¨uredi [22]),

4. m(k2 + k) ≤ k4 + k3 + k2 if a projective plane of order k exists (F¨uredi [22]),

5. m(kn + kn−1) ≤ k2n + k2n−1 + k2n−2 for all n, if a projective plane of order k exists (Drake and Sane [15]),

6. m(k) ≥ 3k if k ≥ 4 (Dow, Drake, F¨uredi and Larson [14],

f(k) 7 7. m(k) < k , where f(k) = c · k 12 (F¨uredi [22]),

8. m(k) ≤ k5 (Blokhuis [3]),

3 2 3 9. m(k) ≤ 4 k + 2 k − 1 if k − 1 is an odd prime power ≥ 7 (Blokhuis [3]),

k2 10. m(k) ≤ 2 + 5k + o(k) if k − 1 is a prime power (Boros, F¨uredi and Kahn [5]). 2.6. SMALL MAXIMAL CLIQUES 55

With those bounds, we can determine m(4): the third bound gives: m(4) ≤ 12, while the sixth gives m(4) ≥ 12. A maximal 4-clique with 12 sets can be constructed as follows: take the sets {0 + i, 1 + i, 4 + i, 6 + i} for i = 0,..., 11 where addition is modulo 12. It is easy to see that any two of those sets intersect. Explicit checking of all cases shows that this clique is maximal. Now consider the q-analogue problem: what is the size (denoted mq(k)) of the smallest maximal intersecting family of rank-k subspaces of some vector space V(n, q). Just as in the set case, where the support of the family was not a parameter of the problem, here the dimension n is not important.

Can we determine mq(2)? We are looking for small maximal intersecting families of rank-2 spaces (projective lines). We saw that the only two possible types of maximal intersecting families of projective lines were point pencils or hyperplanes (all the lines of a hyperplane). Because we can always add more lines to a point pencil, this type will never be maximal. So the smallest maximal intersecting family consists of all the projective lines of a 2 projective plane. Hence mq(2) = q + q + 1. Note that, again, the limit case for q → 1+ is 3 = m(2).

Now what about mq(k) for k ≥ 3?

2 Theorem 2.45. If k − 1 is a prime power, then mq(k) ≤ k − k + 1.

Proof. Let k − 1 be a prime power and consider the maximal k-clique K arising from the lines of the projective plane PG(2, k − 1). We showed that this was a maximal k-clique of size k2 − k + 1. From K we will construct an intersecting family Ke of rank-k spaces, that has the same size, and show that it is maximal.

2 Take a basis {e1, e2, . . . , ek2−k+1} of V(k −k+1, k−1). For each element

K = {i1, i2, . . . , ik} of K, consider the rank-k space Ke =< ei1 , ei2 , . . . , eik > and let Ke be the family of all those spaces. It is obvious that any two of those spaces intersect, so Ke is an intersecting family of k2 − k + 1 rank-k spaces.

Now suppose that Ke is not maximal, that means we can add a rank-k space U that intersects all elements of Ke. Now as each rank-k space, U is the row space of a unique k × (k2 − k + 1) matrix M in row reduced echelon 56 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES form. Without loss of generality we can assume that (maybe after relabeling the basis vectors, and thus relabeling the elements of X) M has this form:

 1 0 ··· 0 ? ··· ?   .   0 1 . ? ··· ?     . .. .   . . . ? ··· ?  0 ······ 1 ? ··· ?

With this labeling, the space < e1, e2, . . . , ek > must be an element of Ke. Suppose that {1, 2, . . . , k} 6∈ K, that means there is a k-set K in K that is disjoint from it, so K ⊆ {k + 1, . . . , k2 − k + 1}. But then U is disjoint from Ke, a contradiction.

Now consider a matrix element Mi,j where 1 ≤ i ≤ k and k + 1 ≤ j ≤ 2 k −k+1. Because K arises from a projective plane, there is a k-set Ki,j ∈ K that intersects {1, 2, . . . , k} in the element i and does not contain the element j. Because U is a blocking set of Ke, U has to intersect Kgi,j. Each vector in this intersection must be a multiple of the i-th row of M, but if Mi,j is a non-zero entry, then none of these vectors belong to Kgi,j. Therefore 2 Mi,j = 0 for each 1 ≤ i ≤ k and k + 1 ≤ j ≤ k − k + 1. That means that U is spanned by the ﬁrst k basis vectors, but then U =< e1, e2, . . . ek >. This is in contradiction with the assumption that U was not an element of Ke.

In the proof of this theorem we constructed a maximal intersecting family of k-spaces from a maximal k-clique arising from a projective plane. We can apply this construction to any maximal k-cliques, also those not arising from a projective plane, but the resulting intersecting family of k-spaces will not always be maximal, as the following example shows.

Example 2.46. The 3-clique consisting of the ten 3-subsets of a 5-set is maximal. If we apply the construction to this clique, we get an intersecting family of 3-spaces. This family is not maximal because (once a basis is chosen) we can add the 3-space < e1 + e2, e3, e4 > for example.

In the proof of Theorem 2.45, we only use the following property of a projective plane: For each line L, for each point p on L and for each point q not on L, there is a line that intersects L in p and does not contain q. We 2.6. SMALL MAXIMAL CLIQUES 57 will call this property in the setting of k-sets in a k-clique property P . So let K be a k-clique with support X, then property P is satisﬁed in K if:

For each k-set K ∈ K, for each x ∈ K and for each y ∈ X \ K there is a k-set K0 ∈ K intersecting K in x and not containing y.

So now we can use the same construction to prove:

Theorem 2.47. If property P is satisﬁed for a maximal k-clique of size m, then there is a maximal intersecting family of k-spaces with the same size. 58 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES Chapter 3

The Kneser and q-Kneser graphs

Besides extremal set theory, extremal graph theory is another important subdomain of extremal combinatorics. A typical problem in here is determining the extremal values of a graph parameter of a certain type of graphs. In this chapter we will look at some graph parameters of the Kneser graph, which we already encountered in Section 1.2.3, and its q-analogue. The ultimate goal is to ﬁnd the chromatic number of the q-Kneser graph.

We already saw that a coloring of a graph is nothing else than a homomorphism of this graph into a complete graph and that in the same way homomorphisms into Kneser graphs are equivalent to multiple colorings. Another reason why the Kneser graph is interesting is because it is an example of a graph with high chromatic number and no short odd cycles. Those objects are much desired in graph theory, but we will come back to this later in this chapter.

3.1 Deﬁnitions and properties

Deﬁnition 3.1 (Kneser graph). The Kneser graph K(n, k), with 1 ≤ k ≤ n, is a graph with vertex set the k-subsets of an n-set, in which two vertices are adjacent if they are disjoint as k-subsets. It is obvious that K(n, k) is an edgeless graph if 2k > n and that K(n, 1) is a complete graph. That is why we will restrict our deﬁnition to k ≥ 2 and 2k ≤ n.

59 60 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS

Without loss of generality we can take {1, 2, . . . , n} for this n-set in the deﬁnition of the Kneser graph and when we write out a vertex as a k-subset we will write the elements in increasing order.

Remark 3.2. • The Johnson scheme J(n, k)(k ≤ n), named after S. M. Johnson, is the association scheme of which the elements are k-subsets of an n-set and the relation between two k-sets (also called the Johnson distance) is half the size of their symmetric distance. For n any 2 ≤ k ≤ 2 , the Kneser graph K(n, k) is the graph that describes the distance-k relation in the Johnson scheme J(n, k). Another graph associated with the Johnson scheme is the Johnson graph, it describes the distance-1 relation in the Johnson scheme.

• Odd graphs are a special case of Kneser graphs: the odd graph Om+1 is the graph with vertices the m-subsets of an 2m + 1-set where two m-subsets are adjacent if and only if they are disjoint, hence Om+1 = K(2m + 1, m).

We have the following properties for the Kneser graph:

Proposition 3.3. For n ≥ 2k we have:

(i) If n > 2k, the automorphism group of K(n, k) is the symmetric group on n symbols, Sym(n);

(ii) K(n, k) is vertex and edge transitive;

n−k (iii) K(n, k) is regular with degree k ;

1 n−kn (iv) K(n, k) has 2 k k edges.

If n = 2k, each vertex has degree 1. It is easy to see that K(2k, k) is 2k 1 2k a graph with k vertices and 2 k edges that are pairwise disjoint (this conﬁguration is sometimes called a perfect matching).

The smallest non-trivial Kneser graph which is not a perfect matching is K(5, 2), which is the Petersen graph (see ﬁgure 3.1).

The q-analogous deﬁnition of the Kneser graphs goes as follows: 3.2. HOMOMORPHISMS. 61

Figure 3.1: K(5, 2) also known as the Petersen graph.

The q-Kneser graph Kq(n, k)(n ≥ 2k) is the graph with vertex set all rank-k subspaces of V(n, q). Two vertices are adjacent if they intersect trivially as subspaces. Or stated projectively: Kq(n, k)(n ≥ 2k) is the graph with vertex set all (k − 1)-spaces of PG(n − 1, q) where two vertices are adjacent if the correspondent subspaces are disjoint.

We have similar properties as those for the Kneser graph: Proposition 3.4. For n > 2k and prime power q we have:

(i) If n > 2k, the automorphism group of Kq(n, k) is isomorphic to PΓL(n, q),

(ii) Kq(n, k) is vertex and edge transitive,

k2 h n−k i (iii) Kq(n, k) is regular with degree q , k q

1 k2 h n−k i n (iv) Kq(n, k) has q edges. 2 k q k q

Here the case n = 2k is not so trivial anymore. We will notice that when we determine the chromatic number.

3.2 Homomorphisms.

In this section we will determine some homomorphisms between Kneser graphs, which will help us to determine the chromatic number later. 62 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS

3.2.1 Homomorphisms between Kneser graphs We give three diﬀerent homomorphisms between Kneser graphs:

Extension map

A k-subset of an n-set is of course also a k-subset of that same n-set with one extra element added. Therefore the vertex set of K(n, k) is a subset of the vertex set of K(n + 1, k). It’s clear that adjacent vertices in K(n, k) are also adjacent in K(n + 1, k) and that all edges in K(n + 1, k) between vertices of K(n, k) are also edges in K(n, k). Therefore K(n, k) is an induced subgraph of K(n + 1, k) and there is a embedding K(n, k) ,→ K(n + 1, k). This homomorphism is called the extension map.

Multiplication map

Take a positive integer t and deﬁne the map µt : K(n, k) → K(tn, tk) as follows: t−1 [ µt(a) = {a1 + in, . . . , ak + in} i=0 for every vertex a = {a1, . . . , ak} of K(n, k). This clearly deﬁnes an embedding which is called the multiplication map.

Coloring map

For K(n, k) with n > 2k, a vertex a = {a1, a2, . . . , ak} is said to be l-regular if al = l and al+1 > l + 1 for some 1 ≤ l ≤ k. A vertex that is not l-regular for any l is called irregular. Because we write the elements of a k-subset in increasing order, l-regularity means that ai = i for 1 ≤ i ≤ l and ai > i for l + 1 ≤ i ≤ k. If a is irregular, this means that ai > i for all i.

In [45] S. Stahl deﬁnes the map η : K(n, k) → K(n − 2, k − 1) as follows:

• if a is l-regular for some l:

η(a) = {a2 − 1, a3 − 1, . . . , al − 1, al+1 − 2, . . . ak − 2}, 3.2. HOMOMORPHISMS. 63

• if a is irregular:

η(a) = {a2 − 2, a3 − 2, . . . , ak − 2}.

This is clearly a map into K(n−2, k −1) but is it also a homomorphism? Take two adjacent vertices a = {a1, a2, . . . , ak} and b = {b1, b2, . . . , bk} of K(n, k). This means that a and b are disjoint as sets, so they cannot both be regular, since then they would both contain the element 1. So we have to check two cases:

1. Both a and b are irregular. In that case η(a) = {a2 −2, a3 −2, . . . , ak − 2} and η(b) = {b2 − 2, b3 − 2, . . . , bk − 2}. If those sets are not disjoint, there is a ai −2 equal to a bj −2 for some 2 ≤ i, j ≤ k, but then ai = bj and the sets were not disjoint, a contradiction. 2. One vertex, say a, is l-regular for some l and the other, b, is irregular. So we have:

η(a) = {a2 − 1, a3 − 1, . . . , al − 1, al+1 − 2, . . . ak − 2},

η(b) = {b2 − 2, b3 − 2, . . . , bk − 2}.

If those sets are not disjoint we have that for some 2 ≤ j ≤ k bj ∈ η(a), so there are two possibilities:

(i) bj − 2 = ai − 2 for some l + 1 ≤ i ≤ k. But then ai = bj and we have a contradiction.

(ii) bj − 2 = ai − 1 for some 2 ≤ i ≤ l. Because 1, 2, . . . , l ∈ a all elements of b must be at least l + 1. Together with bj = ai + 1 = i + 1 ≤ l + 1 this means that bj = l + 1. But l + 1 is the smallest value an element of b can have, so j must be equal to 1, but b1 is not an element of η(b), a contradiction. In each case the assumption that η(a) and η(b) are not disjoint yields a contradiction, so they must be disjoint and hence adjacent as vertices in K(n − 2, k − 1), so η is a homomorphism.

For reasons that will become clear in Section 3.3.1 this homomorphism is called the coloring map.

In the same paper ([45]) Stahl conjectures that any homomorphism from K(n, k) to K(n0, k0) is a composition of extension, multiplication and coloring maps. 64 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS

3.2.2 Homomorphisms between q-Kneser graphs In [9] the following homomorphisms are given:

q-Extension map There is a q analogue of the extension map between ordinary Kneser graphs, this is an embedding Kq(n, k) ,→ Kq(n + 1, k) which is also called the extension map.

q-Multiplication map A rank-k subspace of V(n, qr) for some r ≥ 1 can be viewed as a rank-rk subspace of V(rn, q). This yields an embedding Kqr (n, k) ,→ Kq(rn, rk). This can be seen as a q-analogue of the multiplication map, hence the name q-multiplication map.

Note that there is no q-analogue of the coloring map. We will give an indication of the reason for this in Section 3.3.2. The following two homomorphisms have no analogue for the ordinary Kneser graphs.

Subfield map Because GF(q) is a subfield of GF(qr) for all r ≥ 1, there is a natural homomorphism Kq(n, k) → Kqr (n, k), called the subfield map.

Deletion map Each rank-k space of V(n, q) (once a basis is chosen) is the row space of a unique k × n matrix in row reduced echelon form. When the first line is deleted from this matrix, the first column only contains zeroes, so this column can be deleted too. What is left is a (k − 1) × (n − 1) matrix in row reduced echelon form, with row space a rank-(k − 1) subspace of V(n − 1, q). This defines a map γ : Kq(n, k) → Kq(n − 1, k − 1). The matrices of two adjacent vertices in Kq(n, k) put underneath each other give a 2k ×n matrix of row rank 2k. The matrices of the images under this map underneath each other give a (2k − 2) × (n − 1) matrix. If those images are not adjacent, they 3.3. CHROMATIC NUMBERS 65 are not disjoint as subspaces and the row rank of this (2k−2)×(n−1) matrix is smaller than 2k − 2, but then the row rank of the original 2k × n matrix cannot be 2k. So this map defines a homomorphism, called the deletion map.

3.2.3 A homomorphism from a q-Kneser graph into a Kneser graph n If we forget the vector space structure on V(n, q), this is just a set of 1 q h i rank-1 subspaces and a rank-k subset of V(n, q) is just a k -subset of this 1 q h n i h k i set. This yields an embedding Kq(n, k) ,→ K , . 1 q 1 q

3.3 Chromatic numbers

In this section we will determine the chromatic number of the Kneser graph and present new results on the chromatic number of the q- Kneser graph.

3.3.1 The Kneser graphs We saw that K(2k, k) is a perfect matching. As a consequence the chromatic number is known: χ(K(2k, k)) = 2.

If we write n = 2k + r, we can ﬁnd an upper bound for the chromatic number using the coloring map deﬁned in the previous chapter. By using induction there is a homomorphism

K(2k + r, k) → K(2k + r − 2, k − 1) → ... → K(r + 2, 1).

The last graph in this chain is of course the complete graph Kr+2 so there is a coloring of K(2k + r, k) with r + 2 colors. Hence χ(K(2k + r, k)) ≤ r + 2.

We can easily make this coloring explicit: each vertex that is a k-subset of {1, 2,..., 2k − 1} is given the color 1. Next, all vertices containing the number 2k + r are colored with the color r + 2, all vertices containing the number 2k + r − 1 that are not colored already are given the color r + 1, the uncolored vertices containing 2k+r−2 are colored r, and so on. Finally, the vertices containing the element 2k that are not colored yet are colored with 2. 66 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS

In 1955 M. Kneser [34] conjectured that r + 2 was also a lower bound.

For k = 2 and k = 3 there are nice combinatorial proofs of this conjecture that use the Hilton-Milner theorem (theorem 2.5):

Proposition 3.5. For n ≥ 4 we have that χ(K(n, 2)) = n − 2. Proof. The proof goes by induction on n. For n = 4, K(4, 2) is bipartite, so it has chromatic number 2 = 4 − 2.

Suppose for contradiction that K(n, 2) has a coloring with n − 3 colors for some n > 4. From the Hilton-Milner theorem, it follows that, if a color class contains more than three vertices, the sets lie in a point pencil. If this is the case for some color class, recolor the graph such that this color class forms an entire point pencil. By removing all vertices in this color class we get a coloring of K(n − 1, 2) with n − 4 < (n − 1) − 2 colors, a contradiction with the induction hypothesis. So each color class contains at most three vertices. Therefore, n(n − 1) 3(n − 3) ≥ |K(n, 2)| = . 2 This implies that n2 − 7n + 18 ≤ 0, which is impossible. Hence, the assumption that K(n, 2) can be colored with n − 3 colors is false.

The proof for k = 3 is completely analogous. In that case the Hilton- Milner theorem states that a color class with more than 3n−8 vertices must lie in a point pencil. The calculation in the proof leads to the inequality n3 − 6n2 + 25n − 40 ≤ 0 that has no solutions.

Unfortunately for k ≥ 4 the Hilton-Milner bound does not lead to a contradiction anymore in the method used above, so we need another proof for the general case.

In 1978, L. Lovászproved the general conjecture ([36]) using clever topological constructions and Borsuk’s theorem. This was one of the first and most spectacular applications of topological methods in combinatorics. A few weeks after this I. Bárány [1] found a much shorter proof also using Borsuk’s theorem and a theorem of D. Gale. Here we state Gale’s theorem, Borsuk’s theorem (actually an equivalent statement) and show Bárány’s 3.3. CHROMATIC NUMBERS 67 proof of Kneser’s conjecture.

Theorem 3.6 (D. Gale (1956) [23]). For every m, r ≥ 0 there exists an r r+1 arrangement of 2m + r points on the r-dimensional unit sphere S in R such that every open hemisphere contains at least m of them.

We will call such a set of 2m + r points a Gale set.

d−1 d Two points x and y on the unit sphere S in R are called antipodal if x + y = 0. For an > 0 we call two points -nearly antipodal if ||x + y|| < . d−1 Borsuk’s graph B(d, ) is an inﬁnite graph with vertex set the points of S and two points are adjacent if they are -nearly antipodal.

Borsuk’s theorem is equivalent with:

Theorem 3.7 (K. Borsuk (1933) [6]). For any > 0 the chromatic number of Borsuk’s graph B(d, ) is at least d + 1.

Using this we can now show B´ar´any’s proof of Kneser’s Conjecture:

Theorem 3.8 (Kneser’s Conjecture). The chromatic number of Kneser’s graph K(2k + r, k) (r ≥ 0) is r + 2.

Proof. It is clear that we only have to prove that r + 2 is a lower bound. r Take a Gale set G ⊂ S of 2k + r points. From Gale’s theorem it follows that every open hemisphere contains at least k points of G. It is easy to show that for some δ > 0 this property remains valid if we exclude the rim of width δ of each hemisphere. The√ property thus becomes: there are at r least k points at distance at most 2 − δ from each point in S .

Now take the Kneser graph K(2k + r, k) with vertex set the k-subsets of G and suppose we have a coloring of this graph with c colors. Now we r r construct a coloring of all the points of S . Take a point x ∈√S . There is a k-subset of G in which all the points have distance at most 2 − δ from x. We give this point the same color as the k-subset (as vertex in the Kneser graph) we just mentioned.

To show that this coloring is also a coloring of the Borsuk graph B(r+1, ) for some > 0 we have to show that two -nearly antipodal points have a r diﬀerent color. Take two points x, y ∈ S with the same color. That means that their corresponding k-subsets must intersect (because otherwise they 68 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS are adjacent as vertices in the Kneser graph and√ can not have the same color). So both x and y are at distance at most 2 − δ from a point z in this intersection. If δ is small enough and we take = δ it is impossible that x and y are -nearly antipodal.

Because we have a coloring with c colors of B(r + 1, ), it follows from Theorem 3.7 that c is at least r + 2. After 1987 other proofs of this conjecture appeared (by V.L. Dol’nikov [13, 12] and K.S. Sarkaria [42] among others) but all of them using topological methods. In 2004 J. Matouˇsek[38] gave a purely combinatorial proof (by specializing a combinatorial proof of Tucker’s lemma [47], a topological theorem).

An interesting problem in graph theory is ﬁnding graphs with large chromatic number and some extra constraint. A graph containing a complete graph of size k as a subgraph has chromatic number at least k, but such a graph clearly contains a lot of short cycles. Are there graphs with high chromatic number and no short cycles? It turns out that constructing graphs with high chromatic number and no short even cycles is not so easy. But avoiding short odd cycles is now easier. Suppose we want a graph with chromatic number c, choosing r = c − 2 gives a Kneser graph K(2k + r, k) for each k with this chromatic number. A 3-cycle in a Kneser graph corresponds with three mutually disjoint k-subsets of the (2k + r)-set, which is impossible if 3k > 2k + r. Hence if k > r there are no 3-cycles. Using similar arguments it can be shown that making k even bigger relative to r we can avoid also larger odd cycles, while the chromatic number stays the same.

3.3.2 The q-Kneser graphs The case n > 2k In this case largest independent sets are the families consisting of all rank-k subspaces that have a given rank-1 subspace in common. Such a family has h i size n−1 . A rather easy upper bound here is given as follows: a rank- k−1 q (n − k + 1) subspace has a nontrivial intersection with each rank-k space, so the rank-1 spaces of this subspace color all-k spaces. Therefore:

n − k + 1 χ(Kq(n, k)) ≤ . 1 q 3.3. CHROMATIC NUMBERS 69

For k = 2 A. Chowdhury, C. Godsil and G. Royle [9] prove that this is the correct value:

n − 1 χ(Kq(n, 2)) = . 1 q Remark 3.9. Note that if we use the parameters from the original Kneser graphs this translates to:

r + 3 χ(Kq(4 + r, 2)) = . 1 q Taking the limit for q → 1+ we get r + 3 instead of r + 2. This is a case where the q-analogue does not correspond with the original problem if we take the limit.

What remains to prove is the chromatic number of Kq(2k+r, k) for k > 2 and r > 0. The following lemma shows that if we can prove that for r = 1 the upper bound mentioned above is the right number, we can prove that this upper bound is the chromatic number for all r:

Lemma 3.10. If k + 2 χ(Kq(2k + 1, k)) = 1 q for k ≥ 3 and some values of q, then

n − k + 1 χ(Kq(n, k)) = 1 q for all n > 2k + 1 and the same values of q.

Proof. Using the deletion map, a homomorphism Kq(n, k) → Kq(n−1, k−1), we can construct a homomorphism Kq(2n − 2k − 1, n − k − 1) → ... → Kq(n, k). , Because of our assumption, we have that

n − k + 1 χ(Kq(2(n − k − 1) + 1, n − k − 1)) = . 1 q 70 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS

This, together with the upper bound we get from the coloring described above gives us: n − k + 1 n − k + 1 ≤ χ(Kq(n, k)) ≤ . 1 q 1 q

Now to continue we have to know what is the size of a maximal independent set of Kq(2k + 1, k) that is no point pencil. In the previous chapter we saw that the largest such families (in the case of n = 2k + 1) were q-Hilton- Milner type families, so from Corollary 2.42 we can conclude the following proposition.

Proposition 3.11. The size of a maximal independent set of Kq(2k + 1, k) (k ≥ 3, q ≥ 4) that is no point-pencil is at most cqk2−3 for some positive constant c.

Now we can prove that Kq(2k +1, k) has the asymptotic chromatic number we think it has for k ≥ 3:

Lemma 3.12. For all k ≥ 3, there is a prime power qk such that for all q ≥ qk we have the following. If there is a coloring of Kq(2k + 1, k) with at h i most k+2 colors, then all colors are point pencils. 1 q

h k+2 i Proof. Suppose there is a coloring of Kq(2k + 1, k) with colors. Let 1 q G be the set of centers of the point pencils used in the coloring and B be set of colors that are no point pencil (let’s call those the bad colors), hence h i |G| + |B| ≤ k+2 . 1 q h i There are exactly k+2 (k − 1)-dimensional subspaces that contain a 1 q ﬁxed (k − 2)-dimensional subspace π. If π does not contain a point of G, at most |G| of the (k−1)-spaces containing π are colored by point pencils. This means that at least |B| of the (k − 1)-subspaces containing π are colored with a bad color.

Now count the pairs (π, ρ), where π is a (k − 2)-subspace not containing elements of G, ρ is a (k − 1)-subspace colored by a bad color and ρ contains π. This counting yields: 2k + 1 2 k + 1 |B| ≤ cqk −3|B| . k − 1 q 1 q 3.3. CHROMATIC NUMBERS 71

If |B|= 6 0 we can divide both sides of this inequality by |B|. In that case the left hand side of this inequality is a monic polynomial in q of degree k2 + k − 2 and the right hand side a polynomial of degree k2 + k − 4 with leading coeﬃcient c > 0. That means that for each k ≥ 3 there is a prime power qk such that this inequality does not hold anymore for all q ≥ qk.

In those cases there cannot be any bad colors.

To complete the theorem we need a theorem by R.C. Bose and R.C. Bur- ton of 1966 that says that we need at least an (n − k)-dimensional subspace to color Kq(n, k) with point pencils only. Theorem 3.13 ([7]). If F is a set of points in PG(n − 1, q) that intersects all (k − 1)-subspaces then:

n − k + 1 |F | ≥ . 1 q

Equality holds if, and only if, F is an (n − k)-dimensional subspace.

Now we can prove the chromatic number for (almost) all cases.

Theorem 3.14. For all k ≥ 3 and n > 2k there is a prime power qk such that for all q ≥ qk we have that n − k + 1 χ(Kq(n, k)) = . 1 q

The colors in a minimal coloring of χ(Kq(n, k)) are point pencils with centers on a (n − k)-dimensional subspace.

Proof. Lemma 3.12 together with Theorem 3.13 tells us the chromatic number in the case n = 2k+1. Lemma 3.10 completes the proof of the chromatic number for all n > 2k.

Theorem 3.13 shows that all minimal colorings come from an (n − k)- subspace.

The case n = 2k This case was very easy and not so interesting in the set case. Here on the contrary it is not so straightforward. An independent set in this graph is a collection of rank-k subspaces that pairwise intersect non-trivially. The 72 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS largest independent sets in this graph are the two types described in the q-analogue of the Erd˝os-Ko-Radoproblem: all rank-k subspaces containing a given rank-1 subspace (type I, point pencils) and all rank-k subspaces contained in a hyperplane (type II). We can identify sets of type I with the rank-1 subspace contained in all rank-k subspaces of this set and the sets of type II with the hyperplane that contains all the rank-k subspaces of this h i set. Both types have size 2k−1 and this is also the independence number k−1 q of the q-Kneser graph.

We try to ﬁnd an upper bound for the chromatic number. A color corresponds to an independent set in the graph and by using the largest independent sets to color the graph we get a small chromatic number. An idea is to take a rank-(k + 1) subspace U in V(n, q). Because all the rank-k subspaces intersect this subspace in at least a rank-1 space, taking all the rank-1 spaces (as sets of type I) in this rank-(k + 1) space as colors, all the vertices are colored1. That way we get:

k + 1 α(Kq(2k, k)) = . 1 q Unfortunately, taking the limit for q → 1+ gives k + 1, which is much higher than the chromatic number of the Kneser graph. But we can do better. We still consider the rank-(k+1) space, let us call it U but now we take a rank-k space inside of it, say V. We take qk type I colors: the rank-1 spaces in U that are not in V; and qk−1 type II colors: the hyperplanes that intersect U in V. It is easy to see that all rank-k spaces are colored that way. This gives us:

k k−1 α(Kq(2k, k)) ≤ q + q which, in the limit to the set case, gives us the expected value 2. So maybe we have found the right value?

In a 2001 paper J. Eisfeld, L. Storme and P. Sziklai [16] prove using lengthy counting arguments that for k = 2 this is the chromatic number. In this case we are talking about projective lines in PG(3, q). They also prove that a minimal coloring is of the following form: take a plane and a point on this plane, now take s (with 1 ≤ s ≤ q) lines on that plane through that

1For a vertex meeting U in at least a rank 2 space, we can choose one of the colors in this subspace. 3.3. CHROMATIC NUMBERS 73 point. As type I colors take the points on the plane but not on one of the s lines and as type II colors take the planes that intersect the given plane in one of the s lines.

Until now the only lower bound known in the case k > 2 was the one we get from proposition 1.15:

h 2k i k q k χ(Kq(2k, k)) ≥ = q + 1. h k+1 i 1 q

If we only use point pencils and hyperplanes as colors we can show that you need at least qk + qk−1 colors:

Theorem 3.15. A minimal cover of the rank-k subspaces of V(2k, q), with k ≥ 2, using only point pencils and hyperplanes has size at least qk + qk−1.

Proof. Let P be the set of centers of the point pencils used in the cover and let H be the set of hyperplanes used in the cover. We will show that |P | + |H| ≥ qk + qk−1.

Take a rank-k subspace π that is covered by a point pencil with center a projective point p. We can assume that π is not contained in one of the elements of H, because if each element of P is contained in an element of H then the size of the cover is certainly larger than qk + qk−1 The number of rank-k-subspaces that intersect π in a rank-(k − 1) subspace not containing p is

k + 1 qk−1( − 1). 1 q h i h i Indeed, there are k − k−1 = qk−1 rank-(k − 1) subspaces on π that 1 q 1 q h i do not contain p, and each of those subspaces is contained in k+1 − 1 1 q rank-k subspaces diﬀerent from π.

All those subspaces need to be covered by point pencils or hyperplanes. Take a point pencil with center not on π, this covers at most qk−1 rank-k spaces, one for each rank-(k − 1) space on π not containing p. A hyperplane 74 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS

h k i intersecting π in a rank-(k − 1) space, say πk−1, covers at most rank-k 1 q spaces. But each other hyperplane intersecting π in πk−1 covers at most an additional qk−1 rank-k spaces. That means that the hyperplanes in H cover h i h i at most |H|qk−1 plus an additional qk−1( k − qk−1) = qk−1( k−1 ) of 1 q 1 q those rank-k spaces. So we have:

k − 1 k + 1 (|P | − 1)qk−1 + |H|qk−1 + qk−1 ≥ qk−1( − 1). 1 q 1 q

And this reduces to what we wanted to prove.

For k = 2 this is already the proof for the chromatic number because all maximal independent sets are point pencils and hyperplanes (see Section 2.5.2). For k ≥ 3 there are also other maximal independent sets that are smaller in size, but we do have to worry about them. For k = 3, we can show that the maximal independent sets that are no point pencils or hyperplanes have a size that, as a polynomial in q, has a degree smaller than that of the size of a point pencil or hyperplane (this size 5 6 is 2 q ∼ q :

Lemma 3.16. The size of a maximal independent set of Kq(6, 3) that is no point pencil or hyperplane is at most 6q5.

To show that q3 + q2 really is the chromatic number we need an upper bound on the amount of ‘bad’ colors:

Lemma 3.17. There is a prime power q3 such that for all prime powers 3 2 q ≥ q3 we have the following. In a coloring of Kq(6, 3) with at most q + q colors, the number of colors that are not point pencils or hyperplanes is bounded above by q2 + 6q + 30.

3 2 Proof. Assume we have a coloring of Kq(6, 3) with at most q + q colors. Let G be the set of point pencils and hyperplanes used as a color (we will call these the ’good’ colors) and B be the set of other colors (’bad’ colors). So we have that

|G| + |B| ≤ q3 + q2.

Because those colors have to color all rank-3 spaces we have: 3.3. CHROMATIC NUMBERS 75

6 5 ≤ |G| + |B|6q5 3 q 2 q ! 5 5 ≤ (q3 + q2) − |B| − 6q5 2 q 2 q

Which leads to:

q8 + q7 + q6 + q5 − q3 − q2 − q − 1 |B| ≤ q6 − 5q5 + 2q4 + 2q3 + 2q2 + q + 1 132q5 − 72q4 − 72q3 − 66q2 − 36q − 30 = q2 + 6q + 29 + q6 − 5q5 + 2q4 + 2q3 + 2q2 + q + 1 Since this last fraction becomes strictly smaller than 1 when q is large enough (say q ≥ q3 for some prime power q3) and |B| is a non-negative integer, we have that for q ≥ q3:

|B| ≤ q2 + 6q + 30.

With this upper bound we can prove what we wanted:

Theorem 3.18. There is a positive integer q3 such that for q ≥ q3 the 3 2 chromatic number of the q-Kneser graph Kq(6, 3) is q + q .

3 2 Proof. Assume we have a coloring of Kq(6, 3) with at most q + q colors and consider again the sets G and B as in lemma 3.17. Furthermore, call a rank-3 space colored with a color from G ‘good’ and one colored with a color from B ‘bad’.

Now we will proceed as in theorem 3.15: take a rank-3 space π (this is a projective plane) colored by a point pencil with center the projective point p. Again the number of planes intersecting π in a line not containing p is 2 3 2 q (q + q + q). But now, a number of those planes, say bπ will be ”bad”. If we count the number of those planes colored by an other point pencil and by a hyperplane, we get the same results as in theorem 3.15. This means that:

2 2 3 2 2 (|G| − 1)q ≥ q (q + q + q) − bπ − q (q + 1) 76 CHAPTER 3. THE KNESER AND Q-KNESER GRAPHS

This leads to: q2(q3 + q2 + q) − b − q2(q + 1) |G| ≥ 1 + π q2 b = q3 + q2 − π . q2

2 2 Hence, |B| ≤ bπ/q , or bπ ≥ |B|q . Of course this holds for all planes colored by a point, and all planes colored by a hyperplane (this situation is completely dual). So if we denote b = min{bπ|π is a good plane}, we have that b ≥ |B|q2.

A double counting of the pairs (π, ρ) where π is a good plane, ρ a bad one, and their intersection is a line gives us:

6 b( − |B|6q5) ≤ q(q2 + q + 1)2|B|6q5. 3 q This reduces to

6 |B|( − |B|6q5) ≤ |B|(q2 + q + 1)26q4. 3 q If we assume that |B| > 0 we can divide both sides of this inequality by |B|. For the remaining |B| in this inequality we can use the bound of lemma 3.17 and assume that q is large enough.

Now we see that the left hand side of this equation has degree 9 and the right hand side only degree 8, so if q is large enough then this inequality will be false.

This means that our assumption |B| > 0 is false for q large enough, so in those cases we only have good colors.

We believe the same can be done for k > 3. Chapter 4

A family of point-hyperplane graphs

4.1 Deﬁnition

In the previous chapter we generalized Kneser graphs over projective spaces. This generalization was quite literally: we replaced (sub)sets by (sub)spaces. Two subsets or subspaces were adjacent if they are disjoint. We remarked that this relation was the distance-k relation, or maximal distance relation, in the Johnson scheme J(n, k). In this chapter we will consider a graph with pairs of points and hyperplanes of some projective space as vertices and the adjacency relation will be represented by the maximal distance such two pairs can have.

Deﬁnition 4.1. Deﬁne the graph PH(n + 1, q) as follows: the vertex set is the set of incident point hyperplane pairs of PG(n, q) and two vertices (p, H) and (q, K) are adjacent if p 6∈ K and q 6∈ H.

Note that the discussion in this chapter is inspired by a particular polar q-Kneser graph deﬁned in the next chapter. Indeed, consider the graph Q+ Kq (6, 2) (for a deﬁnition and properties see Chapter 5). It has as vertex set the set L of the totally isotropic projective lines of Q+(5, q) and two lines L, K ∈ L are adjacent in the graph if L ∩ K⊥ = ∅ (or equivalently K ∩ L⊥ = ∅).

Using the Klein correspondence we can give another description of this graph. The set of totally isotropic lines of Q+(5, q) is mapped under the

77 78 CHAPTER 4. A FAMILY OF POINT-HYPERPLANE GRAPHS

Klein correspondence to the set of incident point plane pairs of PG(3, q). For two such pairs (p, H) and (q, K), being at ‘maximal distance’ in the collinearity relation means that p 6∈ K and q 6∈ H. The graphs PH(n + 1, q) are a generalization of this graph over PG(n, q).

Proposition 4.2. The graph PH(n+1, q) as deﬁned above has the following properties: n+1 n (i) PH(n + 1, q) has 1 q 1 q vertices, (ii) PH(n + 1, q) is regular with degree q2n−1. n+1 Proof. (i) There are 1 q hyperplanes in PG(n, q) and each hyperplane n is incident with 1 q points. (ii) Take a vertex (p, H). The hyperplane of an adjacent vertex cannot contain p so there are qn possibilities for this hyperplane. The point of an adjacent vertex cannot lie in H but has to lie in its own hyperplane, so there are qn−1 possibilities.

To ﬁnd the chromatic number of PH(n + 1, q) it is useful to study large cocliques. The following proposition shows that a maximal coclique has a nice “linear” properties:

Proposition 4.3. Let C be a maximal coclique of PH(n + 1, q).

(i) For each hyperplane H of PG(n, q), there is a subspace UH of H such that (p, H) ∈ C if and only if p ∈ UH . Dually, for each point p there is a subspace Up containing p such that (p, H) ∈ C if and only if Up is contained in H.

(ii) Take a hyperplane Y of PG(n, q) and a vertex (p, G) of PH(n, q) de- ﬁned over the projective space Y (so G is an (n − 2)-dimensional projective space contained in Y and p is a point of Y ). There are 0, 1 or q hyperplanes Hi of PG(n, q) such that Hi ∩ Y = G and (p, Hi) ∈ C for all i. Moreover, if it extends in q ways, then (p, Y ) ∈ C.

(iii) Take a hyperplane Y of PG(n, q) and suppose that both vertices (p, G0), (p, G00) of PH(n, q) deﬁned over Y extend in q ways into vertices of C. Then all vertices (p, G) of PH(n, q) for which G0 ∩ G00 ⊆ G extend in q ways into vertices of C. Dually if both vertices (p0,G), (p00,G) of 4.1. DEFINITION 79

PH(n, q) extend in q ways, then all vertices (p, G) for which p ∈ p0p00 extend in q ways.

Proof. (i) Take a hyperplane H and consider the set UH := {p | (p, H) ∈ C}. We have to prove that this set is a subspace of H. This set is clearly a subset of H, so if we take two points p, q ∈ UH and prove that all points on the line pq are in UH we are done. So take a point r on the line pq and suppose (r, H) 6∈ C. This is only possible if there is a vertex (s, K) ∈ C such that both r 6∈ K and s 6∈ H. But because s 6∈ H, both p and q must lie in K, but then pq and hence also r lies in K, a contradiction.

(ii) Denote by Hi, i = 1, . . . , q, the q hyperplanes that, intersected with Y , give G. Suppose (p, G) extends in at least 2 ways into a vertex of C: without loss of generality we can assume that (p, H1), (p, H2) ∈ C. Let Up be the subspace mentioned in (i). Clearly Up ⊆ H1 ∩ H2 = G, so Up ⊆ Hi for i = 3, . . . , q and Up ⊆ Y . Hence (p, Hi) ∈ C for i = 3, . . . , q and (p, Y ) ∈ C.

0 00 (iii) Denote by Hi,H i,H i, i = 1, . . . , q, the q hyperplanes that, intersected with Y , give respectively G, G0 and G00 for a G ⊇ G0 ∩ G00. We 0 00 know that (p, H i), (p, H i) ∈ C for i = 1, . . . , q. Now suppose that (p, H1) 6∈ C, this means that there is a (q, K) ∈ C such that p 6∈ K 0 0 00 00 0 00 and q 6∈ H1. But H 1 ∩ H 2 ∩ H 1 ∩ H 2 = G ∩ G ⊆ G ⊆ H1, so at 0 0 00 00 least one of the four hyperplanes H 1,H 2,H 1,H 2 does not contain q. Hence p ∈ K, a contradiction, so (p, Hi) ∈ C for i = 1, . . . , q.

If we represent a point p of PG(n, q) by a column vector p and a hyperplane H by a row vector H, then a point hyperplane pair (p, H) can be represented by the rank-1 (n + 1) × (n + 1) matrix pH, and this is an incident pair if and only if Hp = 0, which is equivalent with tr(pH) = 0. Now take two vertices x = pH and y = qK, this means that Hp = Kq = 0. Those vertices are adjacent if and only if tr(xy) 6= 0. Indeed, if x and y are non-adjacent then Hq = 0 or Kp = 0, which is equivalent with (Hq)(Kp) = 0. If (Hq)(Kp) = 0 then 0 = tr(HqKp) = tr(pHqK) = tr(xy). If tr(xy) = 0, then 0 = tr(pHqK) = (Hq)tr(pK), which means that Hq = 0 or Kp = tr(pK) = 0, hence x and y are non-adjacent.

Using this representation we can show another linearity property for a maximal coclique: 80 CHAPTER 4. A FAMILY OF POINT-HYPERPLANE GRAPHS

Proposition 4.4. Let C be a maximal coclique of PH(n+1, q). Viewed as a set of matrices C = hCi∩X where X ⊆ sln+1(q) is the set of (n+1)×(n+1) rank 1-matrices with trace 0. Proof. It is obvious that C ⊆ hCi ∩ X. Now take an x ∈ hCi ∩ X and take a basis {x1, . . . , xm} ⊆ C of hCi, so x is a linear combination of the basis elements xi. Because the trace function is linear, tr(x) = 0, so x = pH is a vertex of PH(n + 1, q). Take a vertex y ∈ C. The linearity of the trace function and the fact that tr(xiy) = 0 for each i guarantees that x and y are non-adjacent. This means that x ∈ C and hence hCi ∩ X ⊆ C.

Now take U1,U2 subspaces of PG(n, q) and consider the sets CUi := {(p, H) | p ∈ Ui,Ui ⊆ H} for i = 1, 2. Clearly both sets are cocliques of

PH(n + 1, q) and it is not hard to see that if U1 ⊆ U2 then CU1 ∪ CU2 is also a coclique. Using this fact we can construct a large coclique. Take a chamber (maximal ﬂag) F of PG(n, q). This means that F = {U0,U1,...,Un−1}, where, for each i, Ui is an i-dimensional projective subspace of PG(n, q), and Ui ⊆ Uj if i ≤ j. For this chamber we construct the following set: [ CF := {(p, H) | ∃i ∈ {0, 1 . . . , n − 1} : p ∈ Ui ⊆ H} = CUi . Ui∈F The observation above guarantees that this set is a coclique. One checks that this coclique is maximal. We will call a maximal coclique of this type a chamber-type coclique.

Proposition 4.5. For each maximal ﬂag F of PG(n, q), we have that 2 n−1 d n + 1 |CF | = 1 + 2q + 3q + ... + nq = := fq(n + 1) dn 1 q

Proof. Suppose that F = {U0,U1,...,Un−1}. We prove that for each i = 0, . . . n − 1 the number of incident point hyperplane pairs (p, H) such that i h n−i i p ∈ Ui ⊆ H, and p 6∈ Uj or Uj 6⊆ H for all j < i, is q . Then summing 1 q this up for all i gives f(n). Take an i ∈ {0, 1, . . . n − 1} and a incident point hyperplane (p, H) such that p ∈ Ui ⊆ H. If p ∈ Ui−1, this pair was already counted before, so i p ∈ Ui \ Ui−1. Hence there are q possibilities for p. Each of those points, together with a hyperplane H containing U , is a pair we want to count. h i i h i Now there are n−i such hyperplanes, so in total we have qi n−i of 1 q 1 q pairs. 4.1. DEFINITION 81

The following theorem justiﬁes calling a chamber-type coclique large:

Theorem 4.6. For each n > 0 there is a prime power qn such that for all prime powers q ≥ qn we have that

|C| ≤ fq(n + 1) for all maximal cocliques C of PH(n + 1, q). Proof. Take a maximal coclique C of PH(n + 1, q). We prove by induction that |C| ≤ fq(n + 1).

For n = 1, the vertices of PH(2, q) are just the projective points of PG(1, q) and all vertices are adjacent. A chamber here is also just a point, and so is a chamber-type coclique. Hence |C| ≤ 1 = fq(2) for all maximal cocliques C.

Now take n > 1. Suppose there is a k-dimensional subspace U of PG(n, q) such that C contains all pairs (q, K) for which q ∈ U ⊆ K. In this case there are three types of pairs (p, H) in C: h i h i • p ∈ U ⊆ H: at most k+1 n−k pairs of this type. 1 q 1 q • p 6∈ U: Then U ⊆ H, and we can look at the quotient V(n + 1, q)/U, by mapping (p, H) onto ((p+U)/U, H/U). Those pairs form a coclique C0 in PH(n − k) (the graph deﬁned over the (n − k − 1)-dimensional 0 projective quotient space). By induction we know that |C | ≤ fq(n−k). Now each “point” (p + U)/U in this quotient has qk+1 preimages in PG(n, q), and each “hyperplane” H/U has only one preimage. So we k+1 have at most q fq(n − k) pairs of this type. • U 6⊆ H: Then p ∈ U and we can look at the pairs (p, H ∩ U) in the k-dimensional projective space U. Again those pairs form a coclique 0 0 C , and by induction, |C | ≤ fq(k + 1). Now each “hyperplane” H ∩ U extends in qn−k ways to a hyperplane of PG(n, q) not containing U. n−k So we have at most q fq(k + 1). In total we have that k + 1 n − k k+1 n−k |C| ≤ + q fq(n − k) + q fq(k + 1) = fq(n + 1). 1 q 1 q

Now suppose that there is a subspace U such that for all pairs (p, H) ∈ C 82 CHAPTER 4. A FAMILY OF POINT-HYPERPLANE GRAPHS we have that p ∈ U or U ⊆ H. Take a incident point hyperplane pair (q, K) such that q ∈ U ⊆ K. If q 6∈ H for a pair (p, H) ∈ C then U 6⊆ H, but then p ∈ U ⊆ K, so (q, K) ∈ C and we are in the previous case.

So now we can assume that for all subspaces U of PG(n, q) there is a pair (q, K) ∈ C such that q 6∈ U and U 6⊆ K. Take a basis for the span of C. Because of Proposition 4.3 this is a maximal set of incident point hyperplane pairs {(pi,Hi)|i ∈ I} ⊆ C that are linearly independent as rank-1 matrices. It is clear that |I| ≤ (n + 1)2 − 1. Suppose that the pi’s do not span PG(n, q), then they lie in a hyperplane H. Now take a vertex (q, K) ∈ C. Because of our assumption, there is a vertex (r, L) ∈ C such that r 6∈ K and L 6= K. In terms of matrices this means that the scalar Kr 6= 0. But then ! X X (Hq)(Kr) = H(qK)r = H ai(piHi) r = ai(Hpi)(Hir) = 0 i∈I i∈I for some ai ∈ GF(q), and hence Hq = 0 or q ∈ H. This means that q ∈ H for all (q, K) ∈ C, a contradiction. So the pi must span PG(n, q). In the same way we can prove that the intersection of all the Hi’s is empty.

Now for each pair (p, H) ∈ C we can deﬁne J(p,H) := {j ∈ I|pj ∈ H}. It is clear that J(p,H) 6= I, because otherwise all pi are contained in H, but then they cannot span PG(n, q). So take an i 6∈ J(p,H), this means that pi 6∈ H, but then p ∈ Hi. Now it is clear that J(p,H) is not empty because then p would be in the intersection of all Hi’s and we assumed that this was empty.

Take a vertex (p, H) ∈ C, and deﬁne U := ∩ H . Suppose (p,H) i∈I\J(p,H) i that pj ∈ U(p,H) for each j ∈ J(p,H). Now we can show that U(p,H) is a subspace we assumed did not exist. Indeed take a (q, K) ∈ C such that q 6∈ U(p,H) and U(p,H) 6⊆ K. The ﬁrst condition on (q, K) guarantees that there is an i ∈ I \J(p,H) such that q 6∈ Hi, while the second gives a j ∈ J(p,H) for which pj ∈ U(p,H) \ K. So as a matrix product (Hiq)(Kpj) 6= 0. But ! X X (Hiq)(Kpj) = Hi(qK)pj = Hi ak(pkHk) pj = ak(Hipk)(Hkpj) = 0 k∈I k∈I for some ak ∈ GF(q), because if k ∈ J(p,H), then Hipk = 0, and if k ∈ I \ J(p,H), then Hkpj = 0. So we have q ∈ U(p,H) or U(p,H) ⊆ K for each 4.2. PH(2,Q) 83

(q, K) ∈ C, a contradiction.

So we can assume that for each J ⊆ I, there is a j ∈ J and an i ∈ I \ J such that pj 6∈ Hi. Take a J ⊆ I and the corresponding j ∈ J and i ∈ I \ J such that pj 6∈ Hi, and consider all the pairs (p, H) such that J(p,H) = J. We can map those pairs to the quotient space PG(n, q)/hpji: p is mapped to ppj/hpji and H is mapped to H/hpji. Because i 6∈ J, we have that pi 6∈ H and hence p ∈ Hi. Because pj 6∈ Hi, p is the intersection point of ppj and Hi. This means that the mapping is one to one. In the quotient space there are at most fq(n) vertices in the coclique. Hence for each J ⊆ I there are at most fq(n) vertices in C, so

(n+1)2−1 |C| ≤ 2 fq(n).

For large enough q this is smaller than fq(n + 1).

Let us try to determine the chromatic number for small n:

4.2 PH(2, q)

In this case hyperplanes are points, so the only hyperplane incident with a point, is the point itself. The vertices of the graph PH(1, q) are the points of the projective line PG(1, q), and two vertices are always adjacent. So PH1 is the complete graph Kq+1, and the chromatic number is q + 1.

4.3 PH(3, q)

In the planar case, besides the chamber-type coclique of size 2q + 1, there is another obvious maximal coclique. Take three points p1, p2, p3 not on a line and consider the vertices (p1, p1p2), (p2, p2p3) and (p3, p3p1). This is clearly a coclique and it is trivial to check that it is maximal. This coclique is called the triangular coclique.

Now we can show that there are no other types of maximal cocliques.

Proposition 4.7. If C is a maximal coclique in PH(3, q), then it is of chamber-type or it is a triangular coclique.

Proof. Take a maximal coclique C and an incident point line pair (p, L) contained in C. Let M be another line through p. 84 CHAPTER 4. A FAMILY OF POINT-HYPERPLANE GRAPHS

1. Suppose that (q, M) is in C for a q 6= p. Now take another vertex (r, N) in C. This vertex cannot be adjacent to (p, L) so p ∈ N or r ∈ L.

(a) p ∈ N: now q ∈ N, or if not, r ∈ M. i. M = N: now all other vertices in C must have p as their point or M as their line, so C is of chamber-type. ii. M 6= N: now q 6∈ N, so r = p. Again C can only be of chamber-type with chamber {p, M}. (b) p 6∈ N, so r ∈ L: now q ∈ N, otherwise (q, M) ∼ (r, N), so we have a triangular coclique.

2. Now (p, M) is the only vertex with M as line. We can assume that the same holds for all other lines N through p (except for L), because otherwise consider case 1 with N instead of M. Now take another vertex (q, K) in C, with q 6= p. This vertex has to be non-adjacent with all vertices (p, X) where X is a line through p, so p ∈ K, and we have that C is of chamber-type with chamber {p, K}.

It follows that, in order to ﬁnd a minimal coloring, only those two types of cocliques need to be considered. But we can even do better:

Lemma 4.8. If there is a coloring of PH(3, q) using some triangular cocliques, we can ﬁnd a coloring with the same number of colors that uses only chamber-type cocliques.

Proof. Since each proper subset of a triangular coclique can be colored with a chamber-type coclique, we have that if a triangular coclique cannot be replaced by a chamber-type coclique, all three point-line pairs of the triangular coclique are colored by the triangular coclique only. But that means that also the three point-line pairs in the opposite triangular coclique are colored by that opposite triangular coclique. Now let p be a point that is incident with m lines ppi (for 1 ≤ i ≤ m) such that the point line pairs (p, ppi) are each colored by a diﬀerent triangular coclique. Because our previous argument, m ≥ 2. Now replace each of those m triangular cocliques by the m chamber-type cocliques CFi where Fi{pi, ppi}. This yields a new coloring with the same number of colors and m less triangular cocliques. Now repeat this procedure for each such point p.

Using this lemma, we ﬁnd an upper bound for the chromatic number: 4.3. PH(3,Q) 85

Proposition 4.9. χ(PH(2, q)) ≤ q2 + 1.

Proof. To color this graph we are looking for as little chamber-type cocliques as possible whose union contains all vertices. Look at the incidence graph of PG(2, q). This is the graph with the points and lines as vertices and two vertices are adjacent if and only if they are an incident point line pair. This is a regular bipartite graph of degree q + 1 with q2 + q + 1 vertices in each class. A regular bipartite graph admits perfect matchings (this is a special case of Hall’s theorem). It is clear that a perfect matching of the incidence graph of PG(2, q) is a coloring of PH(2, q) with q2 + q + 1 colors. Of course this coloring is trivial: for all points p take a arbitrary line Lp through it and use the flag-type coclique. But we can do a bit better. Take a flag (p, L) and delete all lines through p and all points on L. In the geometry that is left, all points lie on exactly q lines and all lines contain exactly q points. So the incidence graph of this geometry now is a regular bipartite graph of degree q with q2 vertices in each class. In this case we get a perfect matching (and hence a coloring) of size q2. Together with the flag-type coclique of (p, L) we have a coloring of PH(2, q) with q2 + 1 colors. This gives the upper bound.

Note that for q = 2 this is bound is tight: PH(2, 2) has 21 vertices and a ﬂag-type coclique has size 5, so a coloring must have at least 5 colors. The coloring constructed above has exactly 5 colors, so χ(PH(2, 2)) = 5 = q2 +1.

For q = 4, and actually for all q = r2 for some prime power r, we can do even better. But to do this we need the following deﬁnition and result:

Deﬁnition 4.10. A non-degenerate maximal k-arc of the projective plane PG(2, q) for some 1 ≤ k < q + 1 is a set of points such that each line intersects this set in either 0 or k points. It can be shown that maximal k-arcs can only exist for k a divisor of q.

In 1969 Denniston proved the existence of maximal k-arcs for all divisors k of q if q is even:

Theorem 4.11 (R.H.F Denniston [11]). There exist maximal k-arcs in PG(2, q) for all divisors k of q if q is even.

Using these maximal arcs, we can construct a coloring with less colors if q is an even prime power. 86 CHAPTER 4. A FAMILY OF POINT-HYPERPLANE GRAPHS

Proposition 4.12. If q = r2 for some prime power r, then √ χ(PH(3, q)) ≤ (q + 1)(q + 1 − q). Proof. Take a maximal r-arc A in PG(2, q). It is not hard to count that |A| = r3−r2+r. Indeed take a point on the arc. This point is on q+1 = r2+1 lines that all intersect the arc in r−1 other points. This means that there are (r−1)(r2 −1)+1 = r3 −r2 +r points on A. Now count the pairs (p, L), where p is a point not on the arc and L a line on p that does not intersect the arc, in two ways. We have q2+q+1−r3+r2−r = (r2+1)(r2−r+1) choices for p, and for each p there are r choices for L. Fixing L we have q+1 = r2+1 choices for p. That means that there are exactly r3 − r2 + r lines disjoint from A. Now it is clear that the set of those lines is a dual (0, r)-arc AD: each point lies on either 0 or r lines of this set. Now delete all points of A and all lines of AD. The incidence graph of the resulting geometry is a regular bipartite graph of degree r2 −r +1 with r4 −r3 +2r2 −r +1 vertices in each class. A perfect √ matching gives a coloring with r4 − r3 + 2r2 − r + 1 = q2 + 2q + 1 − (q + 1) q colors, which is the desired result.

The next proposition gives a lower bound for all q and shows that for q = r2 for some prime power r the coloring above is the best possible. Proposition 4.13. √ χ(PH(3, q)) ≥ (q + 1)(q + 1 − q). Proof. Take a coloring of PH(3, q) consisting of chamber-type cocliques only with t colors. That is, we have a collection of incident point line pairs (pi,Li) (1 ≤ i ≤ t) such that for each point line pair (q, K) we have that q = pi for some 1 ≤ i ≤ t or K = Lj for some 1 ≤ j ≤ t (or both). Now let X be the set of points of PG(2, q) that do not occur as pi and Y the set of lines of 2 PG(2, q) that do not occur as Lj. It is clear that |X| = |Y | = q + q + 1 − t. In the incidence graph of PG(2, q) there are no edges between the sets X and Y . A standard interlacing result (see e.g. [25], Corollary 5.3) gives that (q + 1)2|X||Y | ≤ q(q2 + q + 1 − |X|)(q2 + q + 1 − |Y |). This means that (q + 1)2(q2 + q + 1 − t)2 ≤ qt2 or t2 − 2(q + 1)2t + (q + 1)2(q2 + q + 1) ≤ 0. This √ √ inequality results in either t ≥ (q + 1)(q + 1 + q) or t ≥ (q + 1)(q + 1 − q). In both cases we have the bound we tried to prove.

Note that for q = 3 this lower bound gives χ(PH(3, q)) ≥ 10 and the upper bound gives χ(PH(3, q)) ≤ 10. Chapter 5

Polar versions of the q-Kneser graphs

In Chapter 3 we constructed a q-analogue of the Kneser graph using a finite vector space instead of a finite set as a starting point. Of course the construction of a Kneser graph can be generalized to other spaces than just vector spaces. One generalization was considered in Chapter 4. In this chapter we will use (finite) polar spaces as a basis for the construction and look at some variations of the construction.

5.1 Deﬁnition

We deﬁned the q-Kneser graph in Chapter 3 as a generalization of the ordinary Kneser graph by replacing sets by vector spaces (or projective spaces). We can generalize this graph in another way using polar spaces instead of projective spaces. Since the condition for adjacency was disjointness, we can do the same here and deﬁne the polar disjointness graph as follows:

Deﬁnition 5.1. Consider a ﬁnite, rank r, polar space P (n − 1, q) (which has a natural embedding in PG(n−1, q)) with n ≥ 3. The polar disjointness P graph Γq (n, k) associated to the polar space P (n − 1, q) with 1 ≤ k ≤ r is the graph with

• vertex set the totally isotropic (k − 1)-spaces of P (n − 1, q), and

• two vertices are adjacent if their corresponding (k − 1)-spaces are disjoint in P (n − 1, q).

87 88 CHAPTER 5. POLAR VERSIONS OF THE Q-KNESER GRAPHS

For the same reason as before we will only consider polar disjointness graphs with k ≥ 2, because the case k = 1 yields a complete graph. Be- cause 2r ≤ n in P (n − 1, q), and k is bounded from above by r (all maximal subspaces have dimension r − 1), the condition n ≥ 2k to exclude edgeless graphs is satisﬁed automatically.

We can immediately compare the chromatic number of those graphs with the chromatic number of the corresponding q-Kneser graphs.

Proposition 5.2. For all k ≥ 2, n ≥ 2k, prime powers q and projective spaces P (n − 1, q) we have that

P χ(Γq (n, k)) ≤ χ(Kq(n, k)).

Proof. The (k − 1)-spaces of P (n − 1, q) are the totally isotropic (k − 1)- P subspaces of PG(n − 1, q), that means that the vertex set of Γq (n, k) is a subset of the vertex set of Kq(n, k). Two disjoint (k−1)-spaces in P (n−1, q) are also disjoint in PG(n−1, q) and any two disjoint totally isotropic (k−1)- P spaces of PG(n − 1, q) are disjoint in P (n − 1, q). Therefore Γq (n, k) is an induced subgraph of Kq(n, k). Hence the chromatic number of the polar q-Kneser graph is upper bounded by the chromatic number of the q-Kneser graph.

The polar disjointness graphs deﬁned here are a generalization of the q-Kneser graphs over polar spaces. Two vertices of such a graph are adjacent if there corresponding subspaces are disjoint, just as in the q-Kneser graphs. In the q-Kneser graphs over projective spaces two subspaces of a certain rank are disjoint if and only if they lie at maximal distance in the collinearity graph of the subspaces of this rank. In this sense two vertices of the q-Kneser graphs (and also the original Kneser graphs) are adjacent if they are ”at maximal distance” from each other in the vector space (or set respectively).

We use this notion of adjacency to deﬁne the polar q-Kneser graphs:

Deﬁnition 5.3. Consider a ﬁnite, rank r, polar space P (n − 1, q) (which has a natural embedding in PG(n − 1, q)) with n ≥ 3. The polar q-Kneser P graph Kq (n, k) associated to the polar space P (n − 1, q) with 1 ≤ k ≤ r is the graph with

• vertex set the totally isotropic (k − 1)-spaces of P (n − 1, q), and 5.2. CHROMATIC NUMBERS 89

• two vertices U and V and are adjacent if and only if U ∩ V ⊥ = ∅, where V ⊥ is the pole of V (the image of V under the polarity that deﬁnes the polar space P (n − 1, q)).

Note that the condition U ∩ V ⊥ = ∅ is equivalent with V ∩ U ⊥ = ∅. In this case the graph is no longer trivial for k = 1 since all totally isotropic points of a polar space are not necessarily mutually non-orthogonal. The condition n ≥ 2k remains for the same reasons as before.

Remark 5.4. If 2k = n a totally isotropic rank-k subspace is its own pole. In that case the adjacency condition for the polar disjointness graph and the polar q-Kneser graph are the same and hence the two graphs are equal. Here again we can give an immediate upper bound for the chromatic number of these graphs. Proposition 5.5. For all k ≥ 2, n ≥ 2k, prime powers q and projective spaces P (n − 1, q) we have that

P P χ(Kq (n, k)) ≤ χ(Γq (n, k)).

P Proof. By deﬁnition both the polar disjointness graph Γq (n, k) and the polar P q-Kneser graph Kq (n, k) have the same vertex set but two vertices that P are adjacent in Γq (n, k), and hence disjoint in the polar space, are not P P necessarily adjacent in Kq (n, k). Two adjacent vertices in Kq (n, k) are P certainly disjoint as subspaces and hence adjacent in Γq (n, k). Together with Proposition 5.2 this gives the following inequality:

P P χ(Kq (n, k)) ≤ χ(Γq (n, k)) ≤ χ(Kq(n, k)). In the rest of this chapter we will try to determine the chromatic number of some polar disjointness graphs and polar q-Kneser graphs associated to the classical polar spaces. Table 5.1 gives an overview of those classical polar spaces, their rank and the associated polar q-Kneser graph.

5.2 Chromatic numbers

Q+ 5.2.1 Kq (2m + 2, m + 1), m ≥ 2 even, a trivial case There is one case where determining the chromatic number is trivial, namely Q+ Kq (2m + 2, m + 1) with m ≥ 2 even. Note that in this case n = 2k and 90 CHAPTER 5. POLAR VERSIONS OF THE Q-KNESER GRAPHS

P Polar space P (n, q) Rank Kq (n + 1, k) + Q+ Hyperbolic quadric Q (2m + 1, q), m ≥ 1 m + 1 Kq (2m + 2, k) − Q− Elliptic quadric Q (2m + 1, q), m ≥ 2 m Kq (2m + 2, k) Q Parabolic quadric Q(2m, q), m ≥ 2 m Kq (2m + 1, k) W Symplectic space W (2m + 1, q), m ≥ 2 m Kq (2m + 2, k) 2 H Hermitian variety H(2m, q ), m ≥ 2 m Kq2 (2m + 1, k) 2 H Hermitian variety H(2m + 1, q ), m ≥ 1 m + 1 Kq2 (2m + 2, k). Table 5.1: The classical polar spaces and their corresponding polar q-Kneser graphs.

Q+ Q+ hence Kq (2m + 2, m + 1) = Γq (2m + 2, m + 1). The vertices are the generators (maximal totally isotropic m-spaces) of the hyperbolic polar space Q+(2m + 1, q). This polar space has two families of generators. Whether two generators belong to the same family is determined by the dimension of their intersection: two generators belong to diﬀerent families if and only if the parity of the dimension of the intersection of both is equal to the parity of the rank of the polar space (m + 1). In the case that m is even, the rank is odd, so two disjoint generators belong to a diﬀerent family. This means that there are two families of vertices of Q+ Kq (2m + 2, m + 1) and two vertices in the same family cannot have an Q+ edge between them. Therefore Kq (2m + 2, m + 1) is a bipartite graph and:

Q+ χ(Kq (2m + 2, m + 1)) = 2 if m is even.

P 5.2.2 Kq (n, 1) If k = 1, the polar disjointness graph is a complete graph so we only consider P the polar q-Kneser graph Kq (n, 1) here. The vertices of this graph are the totally isotropic points of the polar space P . Two points are adjacent in the graph if they are not perpendicular (in other words, if one point is not in the pole of the other), that means that a coclique is a set of mutual orthogonal points. The largest sets of mutual orthogonal points, and hence the largest cocliques, are the maximal totally isotropic subspaces of the polar space.

Using Proposition 1.15 this already gives a lower bound for the chromatic number. If we could partition the points of the polar space by maximal 5.2. CHROMATIC NUMBERS 91 totally isotropic subspaces, then this lower bound is the chromatic number. Such a partition is called a spread of the polar space:

Deﬁnition 5.6. A spread of a polar space is a partition of the points of the polar space in maximal totally isotropic subspaces.

The following theorem (see e.g. [46]) gives an overview of the existence of spreads in the classical polar spaces:

Theorem 5.7. The following is known for the existence of spreads of classical polar spaces:

1. Q+(2m + 1, q), m ≥ 1:

(a) Q+(2m + 1, q) has no spread if m is even, (b) Q+(2m + 1, q), m odd has a spread if q is even, (c) Q+(3, q) has a spread for all prime powers q, (d) Q+(7, q) has a spread if q is even, q is an odd prime or q ≡ 0 or 2 mod 3.

2. Q−(2m + 1, q), m ≥ 2:

(a) Q−(2m + 1, q) has a spread if q is even, (b) Q−(5, q) has a spread for all q.

3. Q(2m, q), m ≥ 2:

(a) Q(2m, q), q even has spreads for all m ≥ 2, (b) Q(2m, q) with q odd and m even has no spreads, (c) Q(6, q), has a spread if q is even, q is an odd prime or q ≡ 0 or 2 mod 3.

4. W (2m + 1, q), contains a spread for all m ≥ 1 and all prime powers q.

5. H(2m, q2), m ≥ 2: H(4, 4) has no spreads.

6. H(2m + 1, q2), m ≥ 1 has no spreads.

The following cases are still open problems:

• Q+(2m + 1, q), m ≥ 5 odd and q odd,

• Q+(7, q), for q odd, with q ≡ 0 mod 3 and not a prime, 92 CHAPTER 5. POLAR VERSIONS OF THE Q-KNESER GRAPHS

• Q−(2m + 1, q), for q > 2 and q odd,

• Q(2m, q), m ≥ 5 odd and q odd,

• Q(6, q), for q odd, with q ≡ 0 mod 3 and not a prime,

• H(2m, q2) for m > 2,

• H(4, q2) for q > 2.

Let us now look at the chromatic number of each case in more detail.

Q+ 1. Kq (2m + 2, 1), m ≥ 1: This graph has

(qm + 1)(qm+1 − 1) q − 1 vertices. A maximal totally isotropic subspace has rank m + 1, so it m+1 contains 1 q points, and hence

Q+ m χ(Kq (2m + 2, 1)) ≥ q + 1.

Furthermore, by Theorem 5.7 we know that:

Q+ m (a) χ(Kq (2m + 2, 1)) > q + 1 if m is even, Q+ m (b) χ(Kq (2m + 2, 1)) = q + 1 if m is odd and q even, Q+ (c) χ(Kq (4, 1)) = q + 1 for all q, Q+ 3 (d) χ(Kq (8, 1)) = q + 1 if q is even, q is an odd prime or q ≡ 0 or 2 mod 3.

Q− 2. Kq (2m + 2, 1), m ≥ 2: This graph has

(qm − 1)(qm+1 + 1) q − 1 vertices. A maximal totally isotropic subspace has rank m, so it con- m tains 1 q points, and hence

Q− m+1 χ(Kq (2m + 2, 1)) ≥ q + 1.

Furthermore, by Theorem 5.7 we know that 5.2. CHROMATIC NUMBERS 93

Q− m+1 (a) χ(Kq (2m + 2, 1)) = q + 1 if q is even, Q− 3 (b) χ(Kq (6, 1)) = q + 1 for all q.

Q 2m 3. Kq (2m + 1, 1), m ≥ 2: This graph has (q − 1)/(q − 1) vertices. A m maximal totally isotropic subspace has rank m, so it contains 1 q points, and hence

Q m χ(Kq (2m + 1, 1)) ≥ q + 1.

Furthermore, by Theorem 5.7 we know that:

Q m (a) χ(Kq (2m + 1, 1)) = q + 1 if q is even, Q m (b) χ(Kq (2m + 1, 1)) > q + 1 if q is odd and m even, Q 3 (c) χ(Kq (7, 1)) = q + 1 if q is even, q is an odd prime or q ≡ 0 or 2 mod 3.

W 2m+2 4. Kq (2m + 2, 1), m ≥ 2: This graph has 1 q vertices. A maximal m totally isotropic subspace has rank m, so it contains 1 q points, and hence W m χ(Kq (2m + 2, 1)) ≥ q + 1. Furthermore, by Theorem 5.7 we know that in all cases there is a spread, so W m χ(Kq (2m + 2, 1)) = q + 1.

H 5. Kq2 (2m + 1, 1), m ≥ 2: This graph has

(q2m+1 + 1)(q2m − 1) q2 − 1

vertices. A maximal totally isotropic subspace has rank m, so it con- m tains 1 q2 points, and hence

H 2m+1 χ(Kq2 (2m + 1, 1)) ≥ q + 1.

Furthermore, by Theorem 5.7 we know that

H 5 χ(Kq2 (5, 1)) = q + 1. 94 CHAPTER 5. POLAR VERSIONS OF THE Q-KNESER GRAPHS

H 6. Kq2 (2m + 2, 1), m ≥ 1: This graph has

(q2m+2 − 1)(q2m+1 + 1) q2 − 1 vertices. A maximal totally isotropic subspace has rank m + 1, so it m+1 contains 1 q2 points, and hence

H 2m+1 χ(Kq2 (2m + 2, 1)) ≥ q + 1. Furthermore, by Theorem 5.7 we know that

H 2m+1 χ(Kq2 (2m + 2, 1)) > q + 1 for all q and m ≥ 1.

P P 5.2.3 Γq (n, 2) and Kq (n, 2), where P has rank 2 If we consider the deﬁnition of a polar space (Deﬁnition 1.51) for rank 2 we get a point set P, together with a collection B of subspaces of dimension one, the lines, that satisfy the following axioms:

(i) A line, together with the points it contains is (isomorphic to) a projective line.

(ii) The intersection of two lines, or the intersection of a point and a line is again a subspace. In the former case this subspace can be a line, a point or the empty subspace, in the latter case this intersection is a point or the empty subspace.

(iii) Take a line L and a point p not on L. There is exactly one line M that contains p and intersects L in a point.

(iv) There are at least two distinct lines.

The incidence structure described here is a special case of a generalized quadrangle. Indeed, a generalized quadrangle is defined as follows: Definition 5.8. A generalized quadrangle GQ(s, t) (with s, t ≥ 1) is an incidence structure (P, B, I) that satisfies the following axioms: (i) On every line there are exactly s + 1 points. Two distinct lines have at most one point in common.

(ii) On every point there are exactly t + 1 lines. There is at most one line through two distinct points. 5.2. CHROMATIC NUMBERS 95

(iii) For every line L and point p not on L there is exactly one line M and one point q such that p is on M and q is the intersection of L and M.

The numbers s and t are called the parameters of the generalized quadrangles. An easy counting argument shows that a generalized quadrangle GQ(s, t) has (st + 1)(s + 1) points and (st + 1)(t + 1) lines.

Note that since the lines of a polar space of rank 2 are projective lines over GF(q) (or GF(q2) in the Hermitian case) we always have s = q (or s = q2 resp.).

The classical generalized quadrangles are the generalized quadrangles that arise from a classical rank-2 polar space. We list them here, together with their parameters:

• Q+(3, q), the hyperbolic quadric in PG(3, q). Here s = q and t = 1. This is a grid.

• Q(4, q), the parabolic quadric in PG(4, q). Here s = t = q.

• Q−(5, q), the elliptic quadric in PG(5, q). Here s = q and t = q2.

• W (3, q), the symplectic polar space in PG(3, q). Here s = t = q.

• H(3, q2), the Hermitian variety in PG(3, q2). Here s = q2 and t = q.

• H(4, q2), the Hermitian variety in PG(4, q2). There s = q2 and t = q3.

P Let us consider the polar disjointness graphs Γq (n, 2). If the generalized quadrangle is embedded in PG(3, q), the pole of a line is again a line. So P the adjacency condition for both Kq(4, 2)P and Γq (4, 2) are the same, so P both graphs are equal. For the polar disjointness graphs Γq (n, 2) a coclique in the graph is a set of lines in the generalized quadrangle that mutually intersect. Because of axiom (iii), such a set is always a point pencil. That means that the maximal size of a coclique is t + 1 and that the chromatic number is at least st + 1. To color the graph we look for a set of points whose point pencils together contain all lines. Such a set is called a blocking set of the generalized quandrangle. A blocking set is called minimal if the set that remains after removing any point is no longer a blocking set. The size of the smallest minimal blocking set clearly gives the chromatic number of the graph. 96 CHAPTER 5. POLAR VERSIONS OF THE Q-KNESER GRAPHS

If a blocking set partitions the set of all lines into point pencils, this blocking set is also called an ovoid. In other words, an ovoid is a set of points of a generalized quadrangle such that a line on one of the points of this set does not contain any other point of this set. It is clear that if an ovoid exists in a generalized quadrangle, then it is the smallest minimal blocking set of that generalized quadrangle. In this case the chromatic number is exactly st + 1, the size of the ovoid.

The following theorem (see e.g. [46],[40]) gives an overview of the existence of ovoids and in the classical generalized quadrangles. If no ovoids exist, we give the size of the smallest minimal blocking set.

Theorem 5.9. For all prime powers q we have the following:

1. Q+(3, q) has ovoids,

2. Q(4, q) has ovoids,

3. Q−(5, q) has no ovoids. The smallest minimal blocking set in this case has size q3 + q.

4. W (3, q) has ovoids if q is even,

5. W (3, q) has no ovoids if q is odd. The smallest minimal blocking set is not known in this case.

6. H(3, q2) has ovoids,

7. H(4, q2) has no ovoids. The smallest minimal blocking set in this case has size q5 + q2.

This theorem gives us the chromatic numbers of most polar disjointness graphs and some polar q-Kneser graphs for k = 2 and polar rank 2, those numbers are summarized in Table 5.2. The table also shows where the minimal coloring arises from an ovoid (OV) and where it arises from the smallest minimal blocking set (BS). 5.2. CHROMATIC NUMBERS 97

P P P (n − 1, q) Γq (n, 2) χ(Γq (n, 2)) + Q+ Q+ Q (3, q) Γq (4, 2) = Kq (4, 2) q + 1 (OV) − Q− 3 Q (5, q) Γq (6, 2) q + q (BS) Q 2 Q(4, q) Γq (5, 2) q + 1 (OV) W W 2 W (3, q), q even Γq (4, 2) = Kq (4, 2) q + 1 (OV) W W W (3, q), q odd Γq (4, 2) = Kq (4, 2) ? (BS) 2 H 5 2 H(4, q ) Γq2 (5, 2) q + q (BS) 2 H H 3 H(3, q ) Γq2 (4, 2) = Kq2 (4, 2) q + 1 (OV) Table 5.2: The chromatic numbers of the disjointness graphs 98 CHAPTER 5. POLAR VERSIONS OF THE Q-KNESER GRAPHS Chapter 6

Generalized Kneser graphs

In Chapter 3 we defined Kneser graphs and generalized them by defining them over finite projective spaces instead of sets. In Chapter 5 we saw that we could also define Kneser graphs over finite polar spaces. In this chapter we will generalize this idea even further. We will define Chevalley groups and see how it is possible to define Kneser graphs over quotients of Chevalley groups. The graphs defined in Chapters 3 and 5 are special cases of the graphs we will define in this chapter.

6.1 Chevalley groups

6.1.1 Tits systems Deﬁnition 6.1. Let G be a group with a (B,N)-pair. This means that there are subgroups B and N of G with the following properties:

• G is generated by B and N,

• T := B ∩ N is a normal subgroup of N,

• W := N/T is a group generated by a set R of order 2 elements,

• for any r ∈ R and w ∈ W we have that

rBw ⊆ BwB ∪ BrwB

and rBr−1 6⊆ B.

99 100 CHAPTER 6. GENERALIZED KNESER GRAPHS

A 4-tuple (B, N, W, R) with these properties is called a Tits system. The group B of the Tits system (B, N, W, R) is sometimes called the Borel subgroup, T can be called the Cartan subgroup or maximal split torus, and W is called the Weyl group.

Note that from the second property of the deﬁnition it follows that N = NG(T ). If J ⊆ R, we denote by WJ = hJi the subgroup of W generated by J.

A large class of groups having (B,N)-pairs are the groups of Lie type. Definition 6.2. A group G(K) is said to be of Lie type if it is a group of rational points of a reductive linear algebraic group G with values in the field K. In the case K = GF(q), this means that G(K) is a subgroup of a group of invertible matrices over a finite field, hence G ≤ GLn(q) for some n ≥ 2 and prime power q.

So take a group G of Lie type and the corresponding Tits system (B, N, W, R). For w ∈ W the length of w, denoted by l(w), is the length of a shortest expression of w as a product of elements in R. One can prove that (see for example [8], Theorem 10.1.2 (v)) if W is ﬁnite (which is the case here), that there is a unique longest element w0 in W and that element is an involution. Proposition 6.3. For all r ∈ R, w ∈ W we have that: BrwB if l(rw) > l(w), BrBwB = BwB ∪ BrwB otherwise.

Note that if l(rw) > l(w) then l(rw) = l(w) + 1 and otherwise l(rw) = l(w) − 1, because r ∈ R and hence l(r) = 1.

Using this, G can be written as a disjoint union as follows: Theorem 6.4. a G = BwB. w∈W

This decomposition is called the Bruhat decomposition of G. 6.1. CHEVALLEY GROUPS 101

6.1.2 Coxeter systems If G is any group with a Tits system (this includes groups of Lie type), one can prove (see e.g. [8], Theorem 10.5.1 (i)) that (W, R) is a Coxeter system. That is, there is a symmetric matrix M = (mr,t)r,t∈R, the Coxeter matrix, with entries in N ∪ {∞} and diagonal entries 1, such that (rt)mr,t = 1 for all r, t ∈ R.

For this reason, W is also called the Coxeter group of the Tits system. With this Coxeter system, one can associate the Coxeter diagram (R,M). Since M is symmetric this diagram can be viewed as a graph with labeled edges by letting {r, t} for two distinct r, t ∈ R be an edge labeled mr,t whenever mr,t ≥ 3. Labels 3 are usually omitted and edges with label 4 drawn as double edges.

Theorem 6.5. Let (W, R) be a Coxeter system

(i) The choice of the matrix M = (mr,t)r,t∈R associated with (W, R) is uniquely determined by the fact that mr,t is the order of rt for all r, t ∈ R.

(ii) If I ⊆ R, then (WI ,I) is a Coxeter system with WI ∩ R = I. (iii) The system (W, R) is finite if and only if R is finite and every connected component of its Coxeter diagram appears in Figure 6.1 As a corollary any Coxeter system has a decomposition into a direct product of Coxeter groups whose diagrams correspond to the connected components of the diagram of the Coxeter system. A Coxeter system that has a connected diagram is called irreducible. If the Tits system has a Cox- eter system with diagram Xn, the Tits system is said to be of type Xn. Hence, Figure 6.1 gives a complete list of the finite irreducible Coxeter systems.

6.1.3 Root systems Now most of the finite Coxeter systems can be described in terms of root systems. Definition 6.6. A (reduced) root system is a finite collection Φ of nonzero l vectors spanning R , for some l ≥ 1, such that 102 CHAPTER 6. GENERALIZED KNESER GRAPHS

Figure 6.1: The ﬁnite irreducible Coxeter diagrams

• if α ∈ Φ, then Rα ∩ Φ = {α, −α}, 2(β,α) • if α, β ∈ Φ, then (α,α) ∈ Z, and

2(β,α) • if α, β ∈ Φ, then wα(β) := β − (α,α) α ∈ Φ.

It is an easy exercise to see that the wα are involutions. In fact, they are reﬂections in the hyperplane perpendicular to α. Now the group W := hwα|α ∈ Φi is called the Weyl group of the root system.

A root system Φ is called irreducible when it cannot be nontrivially decomposed as Φ1 ∪ Φ2 where all vectors in Φ1 are perpendicular to all vectors in Φ2. It can be shown that each irreducible root system Φ contains a fundamental subsystem of roots ∆.

Deﬁnition 6.7. A fundamental subsystem of roots ∆ of a root system Φ is 6.1. CHEVALLEY GROUPS 103 a subcollection of vectors in Φ such that

l • ∆ is a basis of R , • each root, when written as a linear combination of vectors in ∆, has either only nonnegative or only nonpositive coeﬃcients.

Now one can check that (W, R := {wα|α ∈ ∆}) is a Coxeter system. It is known that each irreducible ﬁnite Coxeter system, except for H3,H4 and m I2 (m = 5 or m ≥ 7), arises in this way. The set R is called the set of fundamental reﬂexions.

6.1.4 Chevalley groups Chevalley groups are a special kind of Lie type groups:

Definition 6.8. The group G = Xl(q) is an untwisted Chevalley group of type Xl over the finite field GF(q) if G is a finite simple group of Lie type, and the following properties hold:

• The Tits system (B, N, W, R) associated to G is of type Xl,

• there is a U C B with B = UT and U ∩ T = {1}, \ • wBw−1 = T . w∈W Because of the second property the Tits system is called split, and because of the third it is called saturated.

The root system of a Chevalley group has a special property: Proposition 6.9. The root system Φ corresponding to the Coxeter system (W, R) of a Chevalley group G has the property that every root α ∈ Φ is an integral combination of the fundamental roots such that either all coeﬃcients are nonnegative (α is called a positive root) or all coeﬃcients are nonpositive (α is called a negative root). We denote by Φ+ and Φ− respectively the sets of positive and negative roots.

Given the root system and the Cartan subgroup T of a Chevalley group, we can ﬁnd back the groups N, W, U and B of the Tits system of G: denote 104 CHAPTER 6. GENERALIZED KNESER GRAPHS by Xα (for α ∈ Φ) the root subgroup with respect to α. Then T normalizes each Xα. Now, because N := NG(T ) and W := N/T , we have that W per- −1 mutes the Xα: wXαw = Xwα for w ∈ W and α ∈ Φ. Now U := Πα∈Φ+ Xα is a subgroup of G normalized by T , so that B := UT is a subgroup of G with B ∩ N = T .

Take w ∈ W , and deﬁne Φw, with ∈ {+, −} as follows: + −1 Φw := {α ∈ Φ |w α ∈ Φ } Using this notation we can deﬁne for w ∈ W and ∈ {+, −}:

Y Uw := Xα. α∈Φw

− + − l(w) Now it can be shown that U = Uw Uw and |Uw | = q .

Proposition 6.10. For a Chevalley group G we have the following decomposition: a − G = Uw wB w∈W −1 + Proof. For a w ∈ W we ﬁrst show that w Uw w ⊆ B. Take an element u + + of Uw , then u = uα1 uα2 ··· uαN , with uαi ∈ Xαi and αi ∈ Φw. This means −1 + −1 −1 that w αi ∈ Φ for each i. Now we have that w Xαi w = Xw αi , hence −1 w w w w w uw ∈ X = X X ··· X = X −1 X −1 ··· X −1 ⊆ U ⊆ B. α1 α2 αN w α1 w α2 w αN Now for w ∈ W we have that BwB = UT wB

= UwT B (because w = nT for some n ∈ N = NG(T )) = UwB (because T ⊆ B) − + = Uw Uw wB − −1 + = Uw ww Uw wB − −1 + = Uw wB (because w Uw w ⊆ B). Substituting this in the Bruhat decomposition a G = BwB. w∈W gives the desired decomposition of G. 6.2. GENERALIZED KNESER GRAPHS 105

Deﬁnition 6.11. A standard parabolic subgroup P of G is a subgroup that contains B.

It can be shown that P = GJ := BWJ B for some J ⊆ R. Using a reasoning as before, one can show that

a − G = Uw wP (6.1) w∈W J

J where W is the set of shortest representatives of left cosets of WJ in W .

6.2 Generalized Kneser graphs

6.2.1 Deﬁnition

For G = Xl(q), a Chevalley group of type Xl over the ﬁeld GF(q), and P , a parabolic subgroup of G, we can deﬁne the generalized Kneser graph Γ over G/P : • The vertex set V Γ of Γ is G/P , that is, the left cosets of P in G, and

• two vertices gP and hP are adjacent if and only if they are opposite, −1 that is, if and only if h g ∈ P w0P , where w0 is the unique longest element of the Coxeter system (W, R). As G acts as a transitive group of automorphisms on Γ, it is clear that this graph is regular.

6.2.2 Parameters To calculate the size of a generalized Kneser graph we can use the following formulas for the size of a Chevalley group and a parabolic subgroup:

+ |G| = q|Φ |(qd1 − 1) ··· (qdl − 1) and + |P | = q|Φ |(q − 1)|R\J|(qe1 − 1) ··· (qem − 1) where the di (i = 1, . . . , l) are the degrees of the diagram of G, and ej (j = 1, . . . , m = |J|) the degrees of the diagram of P . We do not deﬁne here what the degrees of a diagram are but list them in Table 6.1. Note that the formula for |G| is just a special case of the expression for |P | where J = R. 106 CHAPTER 6. GENERALIZED KNESER GRAPHS

Xl d1, d2, . . . , dl Al 2, 3, . . . , l + 1 Bl,Cl 2, 4,..., 2l Dl 2, 4,..., 2l − 2, l E6 2, 5, 6, 8, 9, 12 E7 2, 6, 8, 10, 12, 14, 18 E8 2, 8, 12, 14, 18, 20, 24, 30 F4 2, 6, 8, 12 G2 2, 6

Table 6.1: Degrees of Chevalley groups

Proposition 6.12. If the diagram of G has degrees di (i = 1, . . . , l) and the connected components of the diagram of P are (X1)l1 , (X2)l2 ,..., (Xm)lm , where (Xj)lj has degrees eji (i = 1, . . . , lj), then:

|G| (qd1 − 1) ··· (qdl − 1) |V Γ| = |G/P | = = . (6.2) |P | m Y e ej l (q − 1)|R\J| (q j 1 − 1) ··· (q j − 1) j=1

P P This is a monic polynomial in q of degree di − li + |R \ J|.

The following proposition determines the valency of Γ:

Proposition 6.13. The valency of a generalized Kneser graph Γ over G/P is given by X k = ql(v0) ql(u)

u∈VJ where v0 is the unique word of smallest length in WJ w0WJ and VJ = {u ∈ J WJ |uv0 ∈ W }.

Proof. To calculate the valency of Γ, take the vertex P . Another vertex gP is adjacent with P if g ∈ P w0P . Now {gP |g ∈ P w0P } = {gP |g ∈ P v0P }, where v0 is the unique word of smallest length in WJ w0WJ . This is because there are aJ , bJ ∈ WJ ⊆ P such that w0 = aJ v0bJ , and hence P w0P = −1 −1 −1 −1 P aJ aJ w0bJ bJ P = P aJ w0bJ P = P v0P . Now by Proposition 6.3, 6.2. GENERALIZED KNESER GRAPHS 107

P v0P = BWJ Bv0P

= BWJ v0BP

= BWJ v0P [ = Buv0P

u∈WJ [ − = Uuv0 uv0P. u∈WJ

Note that this union is not a disjoint union. Because of Proposition 6.10 this is a disjoint union if we only consider the u ∈ WJ such that uv0 is right J reduced w.r.t. J. Or in other words, uv0 ∈ W . This means that

|{gP |g ∈ P w0P }| = |{gP |g ∈ P v0P }| a − = | Uuv0 uv0P | u∈VJ X − = |Uuv0 | u∈VJ X = ql(uv0)

u∈VJ X = ql(v0) ql(u)

u∈VJ

J where VJ = {u ∈ WJ |uv0 ∈ W }.

The following proposition shows that a generalised Kneser graph has a large valency compared to its size. As q grows to inﬁnity, the ratio of the size to the valency of the graph tends to 1.

Proposition 6.14. Let Γ be the generalized Kneser graph over G/P , where G is a Chevalley group and P a parabolic subgroup of G. The degree of the size of Γ, as polynomial in q, is the same as the degree of the valency of Γ. Moreover, both polynomials are monic.

Proof. Formula 6.2 uses the degree of the diagrams of G and the degrees the components of the diagram of P to calculate the size of the generalized 108 CHAPTER 6. GENERALIZED KNESER GRAPHS

Kneser graph over G/P , but there is another way of calculating this size. Using the decomposition given by formula 6.1.4 we have that

X − X l(w) |G/P | = |Uw | = q . w∈W J w∈W J The (unique) longest element in W J is clearly the right J reduced form of w0, so the length of this word determines the degree of the size of Γ. On the J other hand, one sees that the unique longest element of W is uv0, where J u is the longest element in WJ such that uv0 ∈ W , and this is also the longest element of VJ , by deﬁnition of VJ . So this uv0 also determines the degree of the valency of Γ. The uniqueness of this element guarantees that both polynomials are monic.

We will see that in a lot of the cases, the valency is actually only one term.

Proposition 6.15. The following statements are equivalent:

(i) J w0 = J,

(ii) J v0 = J,

w0 (iii) WJ = WJ ,

v0 (iv) WJ = WJ .

Theorem 6.16. Let G = Xl(q) be a Chevalley group and P = BWJ B a w parabolic subgroup of G. Then J 0 = J, where w0 is the unique longest word in the Weyl group W , if and only if the valency of the generalized Kneser l(v ) graph Γ over G/P is q 0 , where v0 is the unique word of smallest length in WJ w0WJ .

Proof. Take a u ∈ WJ and consider −1 uv0 = v0v0 uv0 v0 = v0u .

w0 v0 Now if J = J, we have because of Proposition 6.15 that WJ = WJ . This v J v means that u 0 ∈ WJ and hence uv0 ∈ W if and only if u 0 , and hence u, is the empty word. So in that case the valency of Γ is X ql(v0) ql(u) = ql(v0).

u∈VJ 6.2. GENERALIZED KNESER GRAPHS 109

Now let us look in what cases the equality J w0 = J holds:

w0 Al: For a fundamental reﬂection i we have that i = l − i + 1. So if for every i ∈ J (J is symmetric) it holds that l − i + 1 ∈ J we have J w0 = J.

w0 w0 Bl: i = i for all i ∈ {1, 2, . . . , l}, so J = J for all J ⊆ R.

w w D2m: i 0 = i for all i ∈ {1, 2,..., 2m}, so J 0 = J for all J ⊆ R.

w w w D2m+1: i 0 = i for all i ∈ {1, 2,..., 2m − 1} and (2m) 0 = 2m + 1, so J 0 = J for all J ⊆ R such that {2m, 2m + 1} ⊆ J or {2m, 2m + 1} ∩ J = ∅.

w w w w E6: 1 0 = 6, 3 0 = 5, 2 0 = 2 and 4 0 = 4, so if J is symmetric, then J w0 = J.

w w E7: i 0 = i for all i ∈ {1, 2, . . . , l}, so J 0 = J for all J ⊆ R.

w w E8: i 0 = i for all i ∈ {1, 2, . . . , l}, so J 0 = J for all J ⊆ R.

w w F4: i 0 = i for all i ∈ {1, 2, . . . , l}, so J 0 = J for all J ⊆ R.

w w G2: i 0 = i for all i ∈ {1, 2, . . . , l}, so J 0 = J for all J ⊆ R.

We conclude that only in the case of Al, D2m+1 and E6 we can have valencies that are a sum of diﬀerent powers of q, rather than a single power of q.

We start with the cases where J w0 = J. To calculate the size of the generalized Kneser graph, formula 6.2 is used. The valency is just the leading term of the polynomial in q representing the number of vertices. In Appendix A.1 those values are listed for small diagram sizes. As an extra check, we explicitly calculated the valencies in this list to compare them to the corresponding sizes. This calculation was done using the free and open source computer algebra package LiE ([10]). First we have to set the group in which we are working with the command setdefault and then deﬁne the set J (note that sets are not supported by LiE as data sets so we use a vector to represent J). Next we use the built in functions long_word, lr_reduce and length to calculate w0, v0 and l(v0) in each case. The following example calculates the valency of the generalized Kneser graph for G of type A4 and J = {1, 4}: > setdefault A4 > j=[1,4] > w_0=long_word 110 CHAPTER 6. GENERALIZED KNESER GRAPHS

> v_0=lr_reduce(j,w_0,j) > length(v_0) 8

w In the cases where J 0 6= J we need to determine for which u ∈ WJ we J J have that uv0 ∈ W . In LiE it is not so hard to list the elements of W . But we have not found an easy way to list the elements of WJ , so we can use the following proposition:

J Proposition 6.17. Consider the subsets of WJ and W , resp. given by

J VJ = {u ∈ WJ |uv0 ∈ W } and J J −1 V = {x ∈ W |xv0 ∈ WJ }. The map J α : VJ → V , v 7→ vv0 J is a bijection between VJ and V . Proof. The map α is clearly injective. Because for all x ∈ V J we have that −1 −1 J xv0 ∈ WJ and xv0 v0 = x ∈ W the map is also surjective. Hence, X l(u) X l(xv−1) q = q 0 .

u∈VJ x∈V J

The function r_cosets(j) lists the elements of W J ; this is not a built in function but it can be found in a ﬁle distributed with the LiE package: r_cosets(vec s)=for wt row W_orbit(char_v(s)) do print(W_word(wt)) od

Note that some other functions used in the r_cosets function are also found in the same file. We modified this function to, in stead of listing the elements, test if the elements (multiplied by the inverse of v0) are in WJ : val_exp (vec s) = { loc ret = []; loc v = lr_reduce(s,long_word,s); for wt row W_orbit(char_v(s)) do loc u = canonical(W_word(wt)înverse(v)); 6.2. GENERALIZED KNESER GRAPHS 111 if test(u,s) then ret=ret^[length(u)]; fi od; ret } The testing happens in the seventh line: test(u,s), where u and s are both vectors, tests whether each component of u (which represents a Weyl word, each component is an element of R) are equal to at least one component of the vector s (which represents a subset of R): test (int x; vec t) = #test if at least one component of t is #equal to x { loc ans = 0; for i = 1 to size(t) do if !ans && x==t[i] then ans = 1; fi od; ans } test (vec s,t) = #test if each component of s is equal to at least one component #of t { loc ans = 1; for i = 1 to size(s) do if ans && !test(s[i],t) then ans = 0; fi od; ans }

So val_exp(j) returns a vector that has the lengths of all u ∈ VJ as components.

In Tables 6.2 and 6.3 we list the valencies k in all cases (for small parameters) where J w0 6= J. 112 CHAPTER 6. GENERALIZED KNESER GRAPHS

G J P k A2 {1}, {2} A1 q(q + 1) 4 A3 {1}, {3} A1 q (q + 1) 2 {1, 2}, {2, 3} A2 q(q + q + 1) 8 A4 {1}, {2}, {3}, {4} A1 q (q + 1) 4 3 2 {1, 2}, {3, 4} A2 q (q + 2q + 2q + 1) 6 2 {1, 3}, {2, 4} A1A1 q (q + 2q + 1) 3 2 {1, 2, 3}, {2, 3, 4} A3 q(q + q + q + 1) 4 2 {1, 2, 4}, {1, 3, 4} A2A1 q (q + q + 1) 13 A5 {1}, {2}, {4}, {5} A1 q (q + 1) 9 3 2 {1, 2}, {4, 5} A2 q (q + 2q + 2q + 1) 12 {1, 3}, {3, 5} A1A1 q (q + 1) 11 2 {1, 4}, {2, 5} A1A1 q (q + 2q + 1) 10 2 {2, 3}, {3, 4} A2 q (q + q + 1) 4 5 4 3 2 {1, 2, 3}, {3, 4, 5} A3 q (q + 2q + 3q + 3q +2q + 1) 9 2 {1, 2, 4}, {1, 2, 5}, {1, 4, 5}, {2, 4, 5} A2A1 q (q + q + 1) 8 3 2 {1, 3, 4}, {2, 3, 5} A2A1 q (q + 2q + 2q + 1) 4 3 3 {1, 2, 3, 4}, {2, 3, 4, 5} A4 q(q + q + q + q + 1) 4 4 3 2 {1, 2, 3, 5}, {1, 3, 4, 5} A3A1 q (q + q + 2q + q + 1)

w0 Table 6.2: Valencies for G = Al(q)(l ≤ 5) in the case J 6= J. 6.2. GENERALIZED KNESER GRAPHS 113

G J P k 18 D5 {4}, {5} A1 q (q + 1) 17 {1, 4}, {1, 5}, {2, 4}, {2, 5} A1A1 q (q + 1) 15 2 {3, 4}, {3, 5} A2 q (q + q + 1) 15 {1, 2, 4}, {1, 2, 5} A2A1 q (q + 1) 14 2 {1, 3, 4}, {1, 3, 5} A2A1 q (q + q + 1) 11 3 2 {2, 3, 4}, {2, 3, 5} A3 q (q + q + q + 1) 6 4 3 2 {1, 2, 3, 4}, {1, 2, 3, 5} A4 q (q + q + q + q + 1) 34 E6 {1}, {3}, {5}, {6} A1 q (q + 1) 33 {1, 2}, {1, 4}, {2, 3}, A1A1 q (q + 1) {2, 5}, {2, 6}, {4, 6} 30 3 2 {1, 3}, {5, 6} A2 q (q + 2q + 2q + 1) 32 2 {1, 5}, {3, 6} A1A1 q (q + 2q + 1) 31 2 {3, 4}, {4, 5} A2 q (q + q + 1) 29 3 2 {1, 2, 3}, {1, 4, 5}, A2A1 q (q + 2q + 2q + 1) {2, 5, 6}, {3, 4, 6} 31 {1, 2, 4}, {2, 4, 6} A2A1 q (q + 1) 31 2 {1, 2, 5}, {2, 3, 6} A1A1A1 q (q + 2q + 1) 25 5 4 3 2 {1, 3, 4}, {4, 5, 6} A3 q (q + 2q + 3q + 3q +2q + 1) 30 2 {1, 3, 5}, {1, 3, 6}, A2A1 q (q + q + 1) {1, 5, 6}, {3, 5, 6} 27 3 2 {2, 3, 4}, {2, 4, 5} A3 q (q + q + q + 1) 19 7 6 5 4 3 {1, 2, 3, 4}, {2, 4, 5, 6} A4 q (q + 2q + 3q + 4q + 4q +3q2 + 2q + 1) 29 2 {1, 2, 3, 5}, {1, 2, 3, 6}, A2A1A1 q (q + q + 1) {1, 2, 5, 6}, {2, 3, 5, 6} 25 4 3 2 {1, 2, 4, 5}, {2, 3, 4, 6} A3A1 q (q + 2q + 2q + 2q + 1) 22 4 3 2 {1, 3, 4, 5}, {3, 4, 5, 6} A4 q (q + q + q + q + 1) 25 4 3 2 {1, 3, 4, 6}, {1, 4, 5, 6} A3A1 q (q + q + 2q + q + 1) 8 8 7 6 5 4 3 {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6} D5 q (q + q + q + q + 2q + q +q2 + q + 1) 19 6 5 4 3 {1, 2, 3, 4, 6}, {1, 2, 4, 5, 6} A4A1 q (q + q + 2q + 2q +2q2 + q + 1)

w Table 6.3: Valencies for G = D5,E6 in the case J 0 6= J. 114 CHAPTER 6. GENERALIZED KNESER GRAPHS

A similar list with sizes included is found in Appendix A.2.

In the first case (J w0 = J), for a fixed Chevalley group G, the valency of the graph depends only on the size, and hence on the type of P . In the w case where J 0 6= J this is no longer true. For example consider G = E6(q) and J = {1, 2, 3}, J = {1, 2, 4} and J = {1, 3, 5}, respectively. In all three cases P has type A2A1. But in the first case only the A1 component is fixed under w0, in the second case the A2 component is fixed and in the last case, nothing is fixed. So when J w0 6= J, the valency also depends on the action of w0 on the components of the type of P .

6.3 Chromatic numbers in the thin An(1) case

The construction of generalized Kneser graphs can be transferred to the thin case (q = 1). Indeed, take a thin Chevalley group Xl(1). In this case the Weyl group W = hRi of Xl(1) is actually the whole group Xl(1). A parabolic subgroup in this case is of the form P = BhJiB = hJi for some J ⊆ R.

The generalized Kneser graph over Xl(1)/P where P = hJi for some J ⊆ R is the graph with vertex set Xl(1)/P . Two vertices ghJi and hhJi 0 0 are adjacent if and only if g = hjw0j for some j, j ∈ hJi and w0 the unique longest word of W .

Using the same reasoning as in Theorem 6.16 we can show that in the w case where J 0 = J (where w0 is the longest word in the Weyl group) the valency of the generalized Kneser graph is 1. That means that only the other cases are interesting here.

The thin Chevalley group of type An, noted as An(1) is Sym(n + 1), the symmetric group on n + 1 letters, say on the set {1, 2, . . . , n, n + 1}.A parabolic subgroup in this case is a subgroup P = hri1 , . . . , rim i where ri is the transposition (i, i + 1). The unique longest word w0 in this case is the reverse permutation.

Let us consider some examples:

Example 6.18. Take n = 3, so the group is A3(1) = Sym(4) and take P = hr3i. The vertices of this graph are of the form gh(3, 4)i where g ∈ Sym(4). 6.3. CHROMATIC NUMBERS IN THE THIN AN (1) CASE 115

We know that h(3, 4)i must be a stabilizer of the set {gh(3, 4)i | g ∈ Sym(4)}. Therefore we can represent the vertices of this graph as ordered pairs out of {1, 2, 3, 4}, so v = 12. Note that this is also the result of substituting q = 1 in the formula for the valency of this graph in the non-thin case (see Appendix A.2. Two vertices v and v0 in this representation are adjacent in the thin case if there is a w ∈ h(3, 4)iw0h(3, 4)i = h(3, 4)i(1, 4)(2, 3)h(3, 4)i such that vw = v0. This means that k = |h(3, 4)i| = 2. Again substituting q = 1 in the value for k in the non-thin case gives the same result. Take the vertex [1, 2] of this graph. This vertex is adjacent with [1, 2](1,4)(2,3) = [4, 3] and [1, 2](1,4)(2,3)(3,4) = [3, 4] and those two vertices are both also adjacent with [2, 1]. Since k = 2 this results in a 4-cycle. Hence this graph is isomorphic to 3 · C4, the disjoint union of three copies of the cycle with length 4. The chromatic number is trivially 2.

We can generalize this example as follows:

Proposition 6.19. Let Γ be the generalized Kneser graph over An(1)/P for n ≥ 2 and P = hrni. Then (n + 1)! Γ =∼ · C 8 4 and χ(Γ) = 2.

Proof. Take a vertex of Γ, this is an ordered (n−1)-tuple out of {1, 2, . . . , n, n+ 1}. Without loss of generality we can take [1, 2, . . . , n − 1]. This vertex is adjacent with the vertices [n+1, n, n−1,..., 3] and [n, n+1, n−1,..., 3] and those vertices are both also adjacent with [2, 1, . . . , n − 1]. Since the valency is |h(n, n+1)i| = 2 this is a 4-cycle and since this graph is transitive, it must be a disjoint union of 4-cycles. The number of vertices of Γ is

n + 1 (n + 1)! v = (n − 1)! = n − 1 2 so Γ consists of the asserted number of 4-cycles.

Example 6.20. The thin Chevalley group of type A2 is Sym(3), so w0 = (1, 3). Take P = h(2, 3)i. The generalized Kneser graph in this case has as vertex set {1, 2, 3}, so v = 3. It is clear that all elements are adjacent, so this graph is a triangle. Now take A3(1) with P = h(2, 3), (3, 4)i the vertices of this graph are the elements of {1, 2, 3, 4} and all elements are mutually adjacent, so this graph is K4. 116 CHAPTER 6. GENERALIZED KNESER GRAPHS

For general n this gives:

Proposition 6.21. Let Γ be the generalized Kneser graph over An(1)/P for ∼ n ≥ 2 and P = hr2, . . . , rni. Then Γ = Kn+1 and hence χ(Γ) = n + 1. Let us study another example:

Example 6.22. Take A4(1) with P = h(3, 4), (4, 5)i and consider the generalized Kneser graph Γ over A4(1)/P . The vertex set is the set of ordered pairs of {1, 2, 3, 4, 5}, so v = 20. Consider the vertex [1, 2], this vertex is adjacent with the following six vertices: [3, 4], [4, 3], [3, 5], [5, 3], [4, 5] and [5, 4]. Note that those are all the ordered pairs that as a set are disjoint from {1, 2}. This reminds us of the deﬁnition of the original Kneser graph, but instead of pairs we look at ordered pairs.

To continue this example we need some deﬁnitions and results:

Deﬁnition 6.23. Deﬁne the tensor product Γ × Γ0 of two graphs Γ and Γ0 as the graph with vertex set the Cartesian product of the two vertex sets and two vertices (u, u0) and (v, v0) (with u and v vertices of Γ and u0 and v0 vertices of Γ0) are adjacent if and only if u ∼ v in Γ and u0 ∼ v0 in Γ0.

Definition 6.24. We define a loop on a vertex v of a graph Γ as the edge that has v as both endpoints. Note that we didn’t allow this in our original definition of graphs. In what follows we will still hold on to this original definition, so unless stated otherwise graphs have no loops. Now we can consider the complete graph Kn. Take the graph that is the result of adding loops to all vertices of the complete graph. We will denote ◦ this graph by Kn. The following result shows that taking a tensor product of a graph with the complete graph with loops doesn’t change the chromatic number:

Proposition 6.25. For all graphs Γ and all n ≥ 1 we have that

◦ χ(Γ × Kn) = χ(Γ).

Proof. First we will show that each coloring of Γ yields a valid coloring of ◦ Γ × Kn with the same number of colors. Indeed, color each vertex (v, i), ◦ where v is a vertex of Γ and i a vertex of Kn, with the color of v in the original coloring of Γ. To show that this is a valid coloring we take two adjacent vertices (v, i) and (w, j). Now suppose that those vertices are colored with the same color, that means that v and w are colored with the same color. 6.3. CHROMATIC NUMBERS IN THE THIN AN (1) CASE 117

But for (v, i) and (w, j) to be adjacent, v and w have to be adjacent in Γ, but ◦ then they could not have had the same color in Γ. So χ(Γ × Kn) ≤ χ(Γ). ∼ ◦ ◦ Now since Γ = Γ × K1 ≤ Γ × Kn we need at least χ(Γ) colors to color ◦ Γ × Kn. Now let us continue with Example 6.22:

Example 6.26. The vertices of this thin generalized Kneser graph Γ over A4(1)/h(3, 4), (4, 5)i are the ordered pairs out of {1, 2, 3, 4, 5} and two vertices are adjacent if and only if they are disjoint. Consider the Kneser graph K(5, 2) with vertex set only those ordered pairs out of {1, 2, 3, 4, 5} that are in lexicographical order instead of the unordered pairs. This is clearly an induced subgraph of Γ of half the size of Γ. Now we can write a vertex of Γ as ([a, b], σ) := [a, b]σ with a < b and σ ∈ Sym(2), so the vertex set of Γ is the direct product of the vertex set of K(5, 2) and a set with two elements. The condition for adjacency in both Γ and K(5, 2) is the same. One easily checks ∼ ◦ that Γ = K(5, 2)×K2 . Proposition 6.25 tells us that χ(Γ) = χ(K(5, 2)) = 3. Again this example can be generalized:

Proposition 6.27. Let Γ be the generalized Kneser graph over An(1)/P for n ≥ 2 and P = hR \{r1 . . . , ri−1, ri}i = hri+1, . . . , rni for 1 ≤ i ≤ (n + 1)/2, then

∼ ◦ Γ = K(n + 1, i) × Ki! and χ(Γ) = χ(K(n + 1, i)) = n − 2i + 3.

Proof. The vertices of Γ are the ordered i-tuples out of {1, . . . , n, n+1} that are adjacent if they are disjoint. Each vertex can be written as ([t1, . . . , ti], σ) σ := [t1, . . . , ti] , where 1 ≤ t1 < . . . < ti ≤ n + 1 and σ ∈ Sym(i).

Another example shows that we encounter already known cases:

Example 6.28. Let Γ be the generalized Kneser graph over A4(1)/hr1, r3, r4i. In the previous example we saw that if P = hr3, r4i we get ordered pairs (2-tuples) as vertices. Now because r1 = (1, 2) is also in P , a permutation of the ﬁrst two elements of the tuple (in this case there are only two elements in the tuple) does not change the vertex. That means that instead of ordered pairs, the vertices are unordered pairs of {1, 2, 3, 4, 5}. Therefore Γ =∼ K(5, 2). 118 CHAPTER 6. GENERALIZED KNESER GRAPHS

As another example, let ∆ be the generalized Kneser graph A7(1)/hr1, r2, r5, r6, r7i. If P would have been hr5, r6, r7i, the vertex set would have been the set of ordered 4-tuples of {1,..., 8}. Now that r1 and r2 are also in P a permutation of the ﬁrst three elements of the tuple does not change the vertex. Therefore each vertex (4-set) of K(8, 4) only gives rise to 4 vertices of ∆ ∼ ◦ instead of 4!. So ∆ = K(8, 4) × K4 . These examples show how we can describe all generalized q-Kneser graphs over An(1)/P where P = hR \ Si and Q = hSi is of type Al for some (0 ≤ l ≤ n):

Proposition 6.29. Let Γ be the generalized Kneser graph over An(1)/P for n ≥ 2 and

P = hR \{rj, rj+1 . . . , ri−1, ri}i = hr1, r2, . . . , rj−1, ri+1, . . . , rni for some 1 ≤ j ≤ i ≤ n, where ri is the transposition (i, i + 1). (i) If 1 ≤ j ≤ i ≤ (n + 1)/2, then

∼ ◦ Γ = K(n + 1, i) × K i! j! and χ(Γ) = χ(K(n + 1, i)) = n − 2i + 3.

(ii) If (n + 1)/2 < i ≤ n and 1 ≤ j < (n + 1)/2, then ∼ n + 1 ◦ Γ = (2i−n−1)!· K(2(n − i + 1), n − i + 1) × K (n−i+1)! , 2i − n − 1 j!

n+1 which is a disjoint union of 2i−n−1 (2i − n − 1)! copies of the graph ◦ K(2(n − i + 1), n − i + 1) × K (n−i+1)! . In that case j! χ(Γ) = χ(K(2(n − i + 1), n − i + 1)) = 2.

Proof. (i) The vertices of Γ are ordered i-tuples of {1, . . . , n, n + 1} where a permutation of the ﬁrst j doesn’t change the vertex. Two vertices are adjacent if and only if they are disjoint.

(ii) We ﬁrst consider the case where j = 1. We will demonstrate this with an example that completely generalizes. Take for Γ the generalized Kneser graph over A7(1)/hr6, r7i (so n = 7 and i = 5). The vertices 6.3. CHROMATIC NUMBERS IN THE THIN AN (1) CASE 119

are ordered 5-tuples with elements from 1 to 8. Take for example the vertex [1, 2, 3, 4, 5], this is adjacent with the vertex [8, 7, 6, 5, 4] and all vertices obtained by permuting the ﬁrst three (= n − i + 1) elements. All those vertices are also adjacent with the vertices that are obtained by permuting the ﬁrst three elements in [1, 2, 3, 4, 5]. Notice that in all those vertices the last two (= 2i − n − 1) elements are the same (although not necessarily in the same order). In fact two adjacent vertices have the same elements in the last two places but in reversed order. That means that this graph cannot be connected. Now if we delete the last two (= 2i−n−1) elements in all those vertices, then we get the graph with vertices the ordered 3-tuples (where 3 = n − i + 1) of {1, 2, 3, 6, 7, 8} and two vertices are adjacent if they are ◦ disjoint. This is the graph K(6, 3) × K3!, or in general

◦ K(2(n − i + 1), n − i + 1) × K(n−i+1)!.

So we deleted the two (= 2i − n − 1) elements 4 and 5. But we have 8 n+1 2 (= 2i−n−1 ) choices for those elements. And the order in which they appear in the last two places in the i-tuple is important, so for each choice we have another two (= (2i − n − 1)!) possibilities. In this way we get that

n + 1 h i Γ =∼ (2i−n−1)!· K(2(n − i + 1), n − i + 1) × K◦ . 2i − n − 1) (n−i+1)!

Now if j > 1 the same reasoning as before gives the correct result. Note that since j < (n + 1)/2 and (n + 1)/2 < i, we have that j ≤ n − i + 1.

Now let us consider an example where the complement of J is not connected:

Example 6.30. Let Γ be the generalized Kneser graph over A4(1)/hr2, r4i. The vertices of this graph need to be stabilized by P = h(2, 3), (4, 5)i, so they can be represented by a point-pair pair {a, {b, c}}, for distinct a, b, c ∈ {1, 2, 3, 4, 5}. This means that v = 30. Take the vertex {1, {2, 3}}, it is adjacent to the vertices {5, {3, 4}}, {5, {2, 4}}, {4, {3, 5}} and {4, {2, 5}}, so k = 4. Note that if we take a vertex, say {1, {2, 3}} which is determined by the point 1 and the pair {2, 3}, the disjoint pair {4, 5} is also determined. Or in other words, this vertex is also fully determined by the ordered pair [{4, 5}, {2, 3}]. Note that, because of this last representation, 120 CHAPTER 6. GENERALIZED KNESER GRAPHS we can represent the vertices of Γ by the directed edges in the Petersen graph K(5, 2). What is adjacency in this representation? Take the two adjacent vertices {1, {2, 3}} and {5, {3, 4}}, they are represented as [{4, 5}, {2, 3}] and [{1, 2}, {3, 4}]. Note that {4, 5} is adjacent with {1, 2} in K(5, 2) and {2, 3} is not adjacent with {3, 4} in K(5, 2). So two vertices of Γ are adjacent if and only if as directed edges of the Petersen graph, the starting vertices of the directed edges are adjacent and the end vertices of the directed edges are non-adjacent. Now take a maximal coclique in the Petersen graph, this has size 4, and consider the set of vertices of Γ that have their starting vertex in this maximal coclique. This set has size 12 and one can easily check that this is a maximal coclique of Γ. This yields that χ(Γ) ≥ 3. Now each coloring of K(5, 2) yields a coloring of Γ. Indeed, color each vertex of Γ with the color of its starting vertex as a directed graph of the Petersen graph. Take two vertices of Γ and suppose that they have the same color. That means that their starting vertices have the same color in the Petersen graph, but those are adjacent in the Petersen graph, clearly a contradiction. Therefore χ(Γ) = χ(K(5, 2)) = 3. This example generalizes as follows:

Proposition 6.31. Let Γ be the generalized Kneser graph over A2k(1)/P for k ≥ 1 and P = hR \{r1, rk+1}i, then Γ is isomorphic with the graph that has the directed edges of K(2k + 1, k) as vertices, and two vertices are adjacent if and only if their starting vertex are adjacent. Therefore χ(Γ) = χ(K(2k + 1, k)) = 3. Proof. The vertices of Γ are pairs that consist of an element and an unordered k-tuple that does not contain that ﬁrst element. Take a vertex, say {1, {2, 3, . . . , k + 1}}. This vertex is adjacent to all vertices {j, S}, where j ≥ k+2, 1, j 6∈ S, |S| = k and |S ∩{2, . . . , k+2}| = 1. We can represent the vertex {1, {2, 3, . . . , k + 1}} by [{k + 2,..., 2k + 1}, {2, . . . , k + 1}], which is a directed edge of K(2k + 1, k). It is clear that two vertices are adjacent if and only if their starting vertices are adjacent (and hence their end vertices are not). Now take a maximal coclique C of K(2k + 1, k), by Theorem 2.3.1 2k 0 this coclique has size k−1 . The set C consisting of all vertices of Γ that have their starting vertices in C is clearly a maximal coclique of Γ with size 2k k k−1 . This yields that

2k+1 k+1 k k 2k + 1 1 χ(Γ) ≥ = = 2 + . 2k k k k k−1 6.4. CHROMATIC NUMBERS IN THE THICK AN CASE 121

Since χ(Γ) is an integer we have χ(Γ) ≥ 3. Now a coloring of K(2k + 1, k) yields a coloring of Γ. Indeed color each vertex of Γ with the color of its starting vertex in K(2k + 1, k).

6.4 Chromatic numbers in the thick An case

Finally we observe that Proposition 6.29 can be generalized almost literally.

Proposition 6.32. Let Γ be the generalized Kneser graph over An(q)/P where n ≥ 2 and P is the parabolic subgroup P = BhJiB for

J = R \{rj, rj+1 . . . , ri−1, ri} = {r1, r2, . . . , rj−1, ri+1, . . . , rn} for some 1 ≤ j ≤ i ≤ n. (i) If 1 ≤ j ≤ i ≤ (n + 1)/2, then ∼ ◦ Γ = Kq(n + 1, i) × K [i]q! [j]q! and χ(Γ) = χ(Kq(n + 1, i)). (ii) If (n + 1)/2 < i ≤ n and 1 ≤ j < (n + 1)/2, then ∼ n + 1 ◦ Γ = [2i−n−1]q!· Kq(2(n − i + 1), n − i + 1) × K [n−i+1]q! . 2(n − i + 1) q [j]q! In that case

χ(Γ) = χ(Kq(2(n − i + 1), n − i + 1)).

Proof. (i) The vertices of Γ are the ﬂags of PG(n, q) of type (Uj,Uj+1,...,Ui) where Ul (for j ≤ l ≤ i) is an (l − 1)-dimensional subspace of PG(n, q) and two vertices are adjacent if their (i − 1) dimensional subspaces are disjoint. This means that, just as in the thin case, Γ is a tensor product of Kq(n+1, i) and some complete graph with loops. Now each (i − 1)-space in Kq(n + 1, i) gives rise to i i − 1 j + 1 i i − 1 j + 1 [i] ! ... = ... = q i − 1 q i − 2 q j q 1 q 1 q 1 q [j]q! ﬂags of the right type. So this is the number of vertices of the complete graph with loops we are looking for. 122 CHAPTER 6. GENERALIZED KNESER GRAPHS

(ii) Again, the vertices of Γ are the ﬂags of PG(n, q) of type (Uj,Uj+1,...,Ui) where Ul (for j ≤ l ≤ i) is an (l −1)-dimensional subspace of PG(n, q). 0 0 0 But now two vertices (Uj,Uj+1,...,Ui) and (Uj,Uj+1,...,Ui ) are ad- 0 0 jacent if and only if Ul ∩ Ul = ∅ for all j ≤ l ≤ n/2 and Ul ∩ Ul is a 2l − n − 1 space for all n/2 < l ≤ i. 0 0 Now consider the projective space PG(n, q)/(Ui ∩ Ui ). Since Ui ∩ Ui is (2i−n−2)-dimensional, the vertices of whose (i−1)-dimensional sub- 0 space contain Ui ∩Ui in this quotient become ﬂags of PG(2n−2i+1, q) of the type (Vj,...Vn−i+1) and now two vertices are adjacent if and only if their (n − i)-dimensional subspaces are disjoint. So the subgraph of Γ on the vertices whose (i − 1)-dimensional subspace contain 0 Ui ∩ Ui is isomorphic to

◦ Kq(2(n − i + 1), n − i + 1) × K [n−i+1]q! . [j]q!

Now the intersection of the flag (Uj,Uj+1,...,Ui) with a (2i − n − 2)- dimensional subspace contained in Ui is a maximal flag (V1,...V2i−n−1). So for each choice of such a flag we get a copy of this subgraph. Now h n+1 i there are [2i − n − 1]q! ways of choosing such a flag and each 2i−n−1 q choice gives a different copy. Appendix A

Parameters of generalized Kneser graphs

In Chapter 6 we deﬁned generalized Kneser graphs deﬁned over Chevalley groups. We gave a formula for their size and a method of determining their valency. In this appendix we give a list of those parameters for the generalized Kneser graphs over An,Bn,Dn for n ≤ 5 and E6,E7,E8,F4 and G2. Note that for clarity we will denote the ri by i.

A.1 The case J w0 = J

A3 – J = {2}; type of P : A1 v = q5 + 2q4 + 3q3 + 3q2 + 2q + 1 k = q5

A4 – J = {1, 4}; type of P : A1A1 v = q8 + 2q7 + 4q6 + 5q5 + 6q4 + 5q3 + 4q2 + 2q + 1 k = q8

– J = {2, 3}; type of P : A2 v = q7 + 2q6 + 3q5 + 4q4 + 4q3 + 3q2 + 2q + 1 k = q7

A5 – J = {3}; type of P : A1 v = q14 + 4q13 + 10q12 + 19q11 + 30q10 + 41q9 + 49q8 + 52q7 + 49q6 + 41q5 + 30q4 + 19q3 + 10q2 + 4q + 1 k = q14

– J = {1, 5}, {2, 4}; type of P : A1A1 v = q13 + 3q12 + 7q11 + 12q10 + 18q9 + 23q8 + 26q7 + 26q6 + 23q5 + 18q4 + 12q3 + 7q2 + 3q + 1 k = q13

– J = {1, 3, 5}; type of P : A1A1A1 v = q12 + 2q11 + 5q10 + 7q9 + 11q8 + 12q7 + 14q6 + 12q5 + 11q4 + 7q3 + 5q2 + 2q + 1 k = q12

123 124APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

– J = {2, 3, 4}; type of P : A3 v = q9 + 2q8 + 3q7 + 4q6 + 5q5 + 5q4 + 4q3 + 3q2 + 2q + 1 k = q9

– J = {1, 2, 4, 5}; type of P : A2A2 v = q9 + q8 + 2q7 + 3q6 + 3q5 + 3q4 + 3q3 + 2q2 + q + 1 k = q9

B2 – J = {1}, {2}; type of P : A1 v = q3 + q2 + q + 1 k = q3

B3 – J = {1}, {2}, {3}; type of P : A1 v = q8 + 2q7 + 3q6 + 4q5 + 4q4 + 4q3 + 3q2 + 2q + 1 k = q8

– J = {1, 2}; type of P : A2 v = q6 + q5 + q4 + 2q3 + q2 + q + 1 k = q6

– J = {1, 3}; type of P : A1A1 v = q7 + q6 + 2q5 + 2q4 + 2q3 + 2q2 + q + 1 k = q7

– J = {2, 3}; type of P : B2 v = q5 + q4 + q3 + q2 + q + 1 k = q5

B4 – J = {1}, {2}, {3}, {4}; type of P : A1 v = q15 + 3q14 + 6q13 + 10q12 + 14q11 + 18q10 + 21q9 + 23q8 + 23q7 + 21q6 + 18q5 + 14q4 + 10q3 + 6q2 + 3q + 1 k = q15

– J = {1, 2}, {2, 3}; type of P : A2 v = q13 + 2q12 + 3q11 + 5q10 + 6q9 + 7q8 + 8q7 + 8q6 + 7q5 + 6q4 + 5q3 + 3q2 + 2q + 1 k = q13

– J = {1, 3}, {1, 4}, {2, 4}; type of P : A1A1 v = q14 + 2q13 + 4q12 + 6q11 + 8q10 + 10q9 + 11q8 + 12q7 + 11q6 + 10q5 + 8q4 + 6q3 + 4q2 + 2q + 1 k = q14

– J = {3, 4}; type of P : B2 v = q12 + 2q11 + 3q10 + 4q9 + 5q8 + 6q7 + 6q6 + 6q5 + 5q4 + 4q3 + 3q2 + 2q + 1 k = q12

– J = {1, 2, 3}; type of P : A3 v = q10 + q9 + q8 + 2q7 + 2q6 + 2q5 + 2q4 + 2q3 + q2 + q + 1 k = q10

– J = {1, 2, 4}; type of P : A2A1 v = q12 + q11 + 2q10 + 3q9 + 3q8 + 4q7 + 4q6 + 4q5 + 3q4 + 3q3 + 2q2 + q + 1 k = q12

– J = {1, 3, 4}; type of P : B2A1 v = q11 + q10 + 2q9 + 2q8 + 3q7 + 3q6 + 3q5 + 3q4 + 2q3 + 2q2 + q + 1 k = q11

– J = {2, 3, 4}; type of P : B3 v = q7 + q6 + q5 + q4 + q3 + q2 + q + 1 k = q7 A.1. THE CASE J W0 = J 125

B5 – J = {1}, {2}, {3}, {4}, {5}; type of P : A1 v = q24 + 4q23 + 10q22 + 20q21 + 34q20 + 52q19 + 73q18 + 96q17 + 119q16 + 140q15 + 157q14 + 168q13 + 172q12 + 168q11 + 157q10 + 140q9 + 119q8 + 96q7 + 73q6 + 52q5 + 34q4 + 20q3 + 10q2 + 4q + 1 k = q24

– J = {1, 2}, {2, 3}, {3, 4}; type of P : A2 v = q22 + 3q21 + 6q20 + 11q19 + 17q18 + 24q17 + 32q16 + 40q15 + 47q14 + 53q13 + 57q12 + 58q11 + 57q10 + 53q9 + 47q8 + 40q7 + 32q6 + 24q5 + 17q4 + 11q3 + 6q2 + 3q + 1 k = q22

– J = {1, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 5}; type of P : A1A1 v = q23 +3q22 +7q21 +13q20 +21q19 +31q18 +42q17 +54q16 +65q15 +75q14 +82q13 + 86q12 + 86q11 + 82q10 + 75q9 + 65q8 + 54q7 + 42q6 + 31q5 + 21q4 + 13q3 + 7q2 + 3q + 1 k = q23

– J = {4, 5}; type of P : B2 v = q21 + 3q20 + 6q19 + 10q18 + 15q17 + 21q16 + 27q15 + 33q14 + 38q13 + 42q12 + 44q11 + 44q10 + 42q9 + 38q8 + 33q7 + 27q6 + 21q5 + 15q4 + 10q3 + 6q2 + 3q + 1 k = q21

– J = {1, 2, 3}, {2, 3, 4}; type of P : A3 v = q19 + 2q18 + 3q17 + 5q16 + 7q15 + 9q14 + 11q13 + 13q12 + 14q11 + 15q10 + 15q9 + 14q8 + 13q7 + 11q6 + 9q5 + 7q4 + 5q3 + 3q2 + 2q + 1 k = q19

– J = {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {2, 3, 5}; type of P : A2A1 v = q21 + 2q20 + 4q19 + 7q18 + 10q17 + 14q16 + 18q15 + 22q14 + 25q13 + 28q12 + 29q11 + 29q10 + 28q9 + 25q8 + 22q7 + 18q6 + 14q5 + 10q4 + 7q3 + 4q2 + 2q + 1 k = q21

– J = {1, 3, 5}; type of P : A1A1A1 v = q22 + 2q21 + 5q20 + 8q19 + 13q18 + 18q17 + 24q16 + 30q15 + 35q14 + 40q13 + 42q12 + 44q11 + 42q10 + 40q9 + 35q8 + 30q7 + 24q6 + 18q5 + 13q4 + 8q3 + 5q2 + 2q + 1 k = q22

– J = {1, 4, 5}, {2, 4, 5}; type of P : B2A1 v = q20 +2q19 +4q18 +6q17 +9q16 +12q15 +15q14 +18q13 +20q12 +22q11 +22q10 + 22q9 + 20q8 + 18q7 + 15q6 + 12q5 + 9q4 + 6q3 + 4q2 + 2q + 1 k = q20

– J = {3, 4, 5}; type of P : B3 v = q16 + 2q15 + 3q14 + 4q13 + 5q12 + 6q11 + 7q10 + 8q9 + 8q8 + 8q7 + 7q6 + 6q5 + 5q4 + 4q3 + 3q2 + 2q + 1 k = q16

– J = {1, 2, 3, 4}; type of P : A4 v = q15+q14+q13+2q12+2q11+3q10+3q9+3q8+3q7+3q6+3q5+2q4+2q3+q2+q+1 k = q15

– J = {1, 2, 3, 5}; type of P : A3A1 v = q18 + q17 + 2q16 + 3q15 + 4q14 + 5q13 + 6q12 + 7q11 + 7q10 + 8q9 + 7q8 + 7q7 + 6q6 + 5q5 + 4q4 + 3q3 + 2q2 + q + 1 k = q18

– J = {1, 2, 4, 5}; type of P : B2A2 v = q18 + q17 + 2q16 + 3q15 + 4q14 + 5q13 + 6q12 + 7q11 + 7q10 + 8q9 + 7q8 + 7q7 + 6q6 + 5q5 + 4q4 + 3q3 + 2q2 + q + 1 k = q18

– J = {1, 3, 4, 5}; type of P : B3A1 v = q15 + q14 + 2q13 + 2q12 + 3q11 + 3q10 + 4q9 + 4q8 + 4q7 + 4q6 + 3q5 + 3q4 + 2q3 + 2q2 + q + 1 k = q15 126APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

– J = {2, 3, 4, 5}; type of P : B4 v = q9 + q8 + q7 + q6 + q5 + q4 + q3 + q2 + q + 1 k = q9

D4 – J = {1}, {2}, {3}, {4}; type of P : A1 v = q11 + 3q10 + 6q9 + 10q8 + 13q7 + 15q6 + 15q5 + 13q4 + 10q3 + 6q2 + 3q + 1 k = q11

– J = {1, 2}, {2, 3}, {2, 4}; type of P : A2 v = q9 + 2q8 + 3q7 + 5q6 + 5q5 + 5q4 + 5q3 + 3q2 + 2q + 1 k = q9

– J = {1, 3}, {1, 4}, {3, 4}; type of P : A1A1 v = q10 + 2q9 + 4q8 + 6q7 + 7q6 + 8q5 + 7q4 + 6q3 + 4q2 + 2q + 1 k = q10

– J = {1, 2, 3}, {1, 2, 4}, {2, 3, 4}; type of P : A3 v = q6 + q5 + q4 + 2q3 + q2 + q + 1 k = q6

– J = {1, 3, 4}; type of P : A1A1A1 v = q9 + q8 + 3q7 + 3q6 + 4q5 + 4q4 + 3q3 + 3q2 + q + 1 k = q9

D5 – J = {1}, {2}, {3}; type of P : A1 v = q19 + 4q18 + 10q17 + 20q16 + 34q15 + 51q14 + 69q13 + 86q12 + 99q11 + 106q10 + 106q9 + 99q8 + 86q7 + 69q6 + 51q5 + 34q4 + 20q3 + 10q2 + 4q + 1 k = q19

– J = {1, 2}, {2, 3}; type of P : A2 v = q17 + 3q16 + 6q15 + 11q14 + 17q13 + 23q12 + 29q11 + 34q10 + 36q9 + 36q8 + 34q7 + 29q6 + 23q5 + 17q4 + 11q3 + 6q2 + 3q + 1 k = q17

– J = {1, 3}, {4, 5}; type of P : A1A1 v = q18 + 3q17 + 7q16 + 13q15 + 21q14 + 30q13 + 39q12 + 47q11 + 52q10 + 54q9 + 52q8 + 47q7 + 39q6 + 30q5 + 21q4 + 13q3 + 7q2 + 3q + 1 k = q18

– J = {1, 2, 3}, {3, 4, 5}; type of P : A3 v = q14+2q13+3q12+5q11+7q10+8q9+9q8+10q7+9q6+8q5+7q4+5q3+3q2+2q+1 k = q14

– J = {1, 4, 5}, {2, 4, 5}; type of P : A1A1A1 v = q17 + 2q16 + 5q15 + 8q14 + 13q13 + 17q12 + 22q11 + 25q10 + 27q9 + 27q8 + 25q7 + 22q6 + 17q5 + 13q4 + 8q3 + 5q2 + 2q + 1 k = q17

– J = {1, 2, 4, 5}; type of P : A2A1A1 v = q15 + q14 + 3q13 + 4q12 + 6q11 + 7q10 + 9q9 + 9q8 + 9q7 + 9q6 + 7q5 + 6q4 + 4q3 + 3q2 + q + 1 k = q15

– J = {1, 3, 4, 5}; type of P : A3A1 v = q13 + q12 + 2q11 + 3q10 + 4q9 + 4q8 + 5q7 + 5q6 + 4q5 + 4q4 + 3q3 + 2q2 + q + 1 k = q13

– J = {2, 3, 4, 5}; type of P : D4 v = q8 + q7 + q6 + q5 + 2q4 + q3 + q2 + q + 1 k = q8

E6 – J = {2}, {4}; type of P : A1 v = q35 +5q34 +15q33 +35q32 +70q31 +125q30 +204q29 +310q28 +444q27 +604q26 + A.1. THE CASE J W0 = J 127

785q25 + 980q24 + 1179q23 + 1370q22 + 1541q21 + 1681q20 + 1780q19 + 1831q18 + 1831q17 + 1780q16 + 1681q15 + 1541q14 + 1370q13 + 1179q12 + 980q11 + 785q10 + 604q9 + 444q8 + 310q7 + 204q6 + 125q5 + 70q4 + 35q3 + 15q2 + 5q + 1 k = q35

– J = {1, 6}, {3, 5}; type of P : A1,A1 v = q34 +4q33 +11q32 +24q31 +46q30 +79q29 +125q28 +185q27 +259q26 +345q25 + 440q24 + 540q23 + 639q22 + 731q21 + 810q20 + 871q19 + 909q18 + 922q17 + 909q16 + 871q15 + 810q14 + 731q13 + 639q12 + 540q11 + 440q10 + 345q9 + 259q8 + 185q7 + 125q6 + 79q5 + 46q4 + 24q3 + 11q2 + 4q + 1 k = q34

– J = {2, 4}; type of P : A2 v = q33 +4q32 +10q31 +21q30 +39q29 +65q28 +100q27 +145q26 +199q25 +260q24 + 326q23 + 394q22 + 459q21 + 517q20 + 565q19 + 599q18 + 616q17 + 616q16 + 599q15 + 565q14 +517q13 +459q12 +394q11 +326q10 +260q9 +199q8 +145q7 +100q6 +65q5 + 39q4 + 21q3 + 10q2 + 4q + 1 k = q33

– J = {1, 2, 6}, {1, 4, 6}, {2, 3, 5}; type of P : A1A1A1 v = q33 + 3q32 + 8q31 + 16q30 + 30q29 + 49q28 + 76q27 + 109q26 + 150q25 + 195q24 + 245q23 + 295q22 + 344q21 + 387q20 + 423q19 + 448q18 + 461q17 + 461q16 + 448q15 + 423q14 + 387q13 + 344q12 + 295q11 + 245q10 + 195q9 + 150q8 + 109q7 + 76q6 + 49q5 + 30q4 + 16q3 + 8q2 + 3q + 1 k = q33

– J = {3, 4, 5}; type of P : A3 v = q30 + 3q29 + 6q28 + 11q27 + 19q26 + 29q25 + 41q24 + 56q23 + 73q22 + 90q21 + 107q20 + 124q19 + 138q18 + 148q17 + 155q16 + 158q15 + 155q14 + 148q13 + 138q12 + 124q11 + 107q10 + 90q9 + 73q8 + 56q7 + 41q6 + 29q5 + 19q4 + 11q3 + 6q2 + 3q + 1 k = q30

– J = {1, 2, 4, 6}; type of P : A2A1A1 v = q31 + 2q30 + 5q29 + 9q28 + 16q27 + 24q26 + 36q25 + 49q24 + 65q23 + 81q22 + 99q21 + 115q20 + 130q19 + 142q18 + 151q17 + 155q16 + 155q15 + 151q14 + 142q13 + 130q12 +115q11 +99q10 +81q9 +65q8 +49q7 +36q6 +24q5 +16q4 +9q3 +5q2 +2q +1 k = q31

– J = {1, 3, 5, 6}; type of P : A2A2 v = q30 + 2q29 + 4q28 + 8q27 + 13q26 + 19q25 + 28q24 + 38q23 + 48q22 + 60q21 + 72q20 + 82q19 + 91q18 + 99q17 + 103q16 + 104q15 + 103q14 + 99q13 + 91q12 + 82q11 + 72q10 + 60q9 + 48q8 + 38q7 + 28q6 + 19q5 + 13q4 + 8q3 + 4q2 + 2q + 1 k = q30

– J = {2, 3, 4, 5}; type of P : D4 v = q24 + 2q23 + 3q22 + 4q21 + 7q20 + 9q19 + 11q18 + 13q17 + 17q16 + 18q15 + 19q14 + 20q13 +22q12 +20q11 +19q10 +18q9 +17q8 +13q7 +11q6 +9q5 +7q4 +4q3 +3q2 +2q+1 k = q24

– J = {1, 2, 3, 5, 6}; type of P : A2A2A1 v = q29 + q28 + 3q27 + 5q26 + 8q25 + 11q24 + 17q23 + 21q22 + 27q21 + 33q20 + 39q19 + 43q18 + 48q17 + 51q16 + 52q15 + 52q14 + 51q13 + 48q12 + 43q11 + 39q10 + 33q9 + 27q8 + 21q7 + 17q6 + 11q5 + 8q4 + 5q3 + 3q2 + q + 1 k = q29

– J = {1, 3, 4, 5, 6}; type of P : A5 v = q21 + q20 + q19 + 2q18 + 3q17 + 3q16 + 4q15 + 5q14 + 5q13 + 5q12 + 6q11 + 6q10 + 5q9 + 5q8 + 5q7 + 4q6 + 3q5 + 3q4 + 2q3 + q2 + q + 1 k = q21

E7 – J = {1}, {2}, {3}, {4}, {5}, {6}, {7}; type of P : v = q62 + 6q61 + 21q60 + 56q59 + 126q58 + 252q57 + 461q56 + 786q55 + 1265q54 + 128APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

1940q53 + 2855q52 + 4054q51 + 5578q50 + 7462q49 + 9732q48 + 12402q47 + 15472q46 + 18926q45 + 22731q44 + 26836q43 + 31173q42 + 35658q41 + 40194q40 + 44674q39 + 48985q38 + 53012q37 + 56643q36 + 59774q35 + 62313q34 + 64184q33 + 65330q32 + 65716q31 + 65330q30 + 64184q29 + 62313q28 + 59774q27 + 56643q26 + 53012q25 + 48985q24 + 44674q23 + 40194q22 + 35658q21 + 31173q20 + 26836q19 + 22731q18 + 18926q17 +15472q16 +12402q15 +9732q14 +7462q13 +5578q12 +4054q11 +2855q10 + 1940q9 + 1265q8 + 786q7 + 461q6 + 252q5 + 126q4 + 56q3 + 21q2 + 6q + 1 k = q62 – J = {1, 2}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 5}, {2, 6}, {2, 7}, {3, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 7}; type of P : A1A1 v = q61 + 5q60 + 16q59 + 40q58 + 86q57 + 166q56 + 295q55 + 491q54 + 774q53 + 1166q52 + 1689q51 + 2365q50 + 3213q49 + 4249q48 + 5483q47 + 6919q46 + 8553q45 + 10373q44 + 12358q43 + 14478q42 + 16695q41 + 18963q40 + 21231q39 + 23443q38 + 25542q37 + 27470q36 + 29173q35 + 30601q34 + 31712q33 + 32472q32 + 32858q31 + 32858q30 + 32472q29 + 31712q28 + 30601q27 + 29173q26 + 27470q25 + 25542q24 + 23443q23 + 21231q22 + 18963q21 + 16695q20 + 14478q19 + 12358q18 + 10373q17 + 8553q16 + 6919q15 + 5483q14 + 4249q13 + 3213q12 + 2365q11 + 1689q10 + 1166q9 + 774q8 + 491q7 + 295q6 + 166q5 + 86q4 + 40q3 + 16q2 + 5q + 1 k = q61

– J = {1, 3}, {2, 4}, {3, 4}, {4, 5}, {5, 6}, {6, 7}; type of P : A2 v = q60 +5q59 +15q58 +36q57 +75q56 +141q55 +245q54 +400q53 +620q52 +920q51 + 1315q50 + 1819q49 + 2444q48 + 3199q47 + 4089q46 + 5114q45 + 6269q44 + 7543q43 + 8919q42 + 10374q41 + 11880q40 + 13404q39 + 14910q38 + 16360q37 + 17715q36 + 18937q35 + 19991q34 + 20846q33 + 21476q32 + 21862q31 + 21992q30 + 21862q29 + 21476q28 + 20846q27 + 19991q26 + 18937q25 + 17715q24 + 16360q23 + 14910q22 + 13404q21 +11880q20 +10374q19 +8919q18 +7543q17 +6269q16 +5114q15 +4089q14 + 3199q13 + 2444q12 + 1819q11 + 1315q10 + 920q9 + 620q8 + 400q7 + 245q6 + 141q5 + 75q4 + 36q3 + 15q2 + 5q + 1 k = q60 – J = {1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {1, 3, 6}, {1, 3, 7}, {1, 4, 5}, {1, 5, 6}, {1, 6, 7}, {2, 4, 6}, {2, 4, 7}, {2, 5, 6}, {2, 6, 7}, {3, 4, 6}, {3, 4, 7}, {3, 5, 6}, {3, 6, 7}, {4, 5, 7}, {4, 6, 7}; type of P : A2A1 v = q59 +4q58 +11q57 +25q56 +50q55 +91q54 +154q53 +246q52 +374q51 +546q50 + 769q49 + 1050q48 + 1394q47 + 1805q46 + 2284q45 + 2830q44 + 3439q43 + 4104q42 + 4815q41 + 5559q40 + 6321q39 + 7083q38 + 7827q37 + 8533q36 + 9182q35 + 9755q34 + 10236q33 + 10610q32 + 10866q31 + 10996q30 + 10996q29 + 10866q28 + 10610q27 + 10236q26 + 9755q25 + 9182q24 + 8533q23 + 7827q22 + 7083q21 + 6321q20 + 5559q19 + 4815q18 + 4104q17 + 3439q16 + 2830q15 + 2284q14 + 1805q13 + 1394q12 + 1050q11 + 769q10 + 546q9 + 374q8 + 246q7 + 154q6 + 91q5 + 50q4 + 25q3 + 11q2 + 4q + 1 k = q59 – J = {1, 2, 5}, {1, 2, 6}, {1, 2, 7}, {1, 4, 6}, {1, 4, 7}, {1, 5, 7}, {2, 3, 5}, {2, 3, 6}, {2, 3, 7}, {2, 5, 7}, {3, 5, 7}; type of P : A1A1A1 v = q60 + 4q59 + 12q58 + 28q57 + 58q56 + 108q55 + 187q54 + 304q53 + 470q52 + 696q51 + 993q50 + 1372q49 + 1841q48 + 2408q47 + 3075q46 + 3844q45 + 4709q44 + 5664q43 +6694q42 +7784q41 +8911q40 +10052q39 +11179q38 +12264q37 +13278q36 + 14192q35 + 14981q34 + 15620q33 + 16092q32 + 16380q31 + 16478q30 + 16380q29 + 16092q28 + 15620q27 + 14981q26 + 14192q25 + 13278q24 + 12264q23 + 11179q22 + 10052q21 + 8911q20 + 7784q19 + 6694q18 + 5664q17 + 4709q16 + 3844q15 + 3075q14 + 2408q13 + 1841q12 + 1372q11 + 993q10 + 696q9 + 470q8 + 304q7 + 187q6 + 108q5 + 58q4 + 28q3 + 12q2 + 4q + 1 k = q60

– J = {1, 3, 4}, {2, 3, 4}, {2, 4, 5}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}; type of P : A3 v = q57 +4q56 +10q55 +21q54 +40q53 +70q52 +114q51 +176q50 +260q49 +370q48 + 509q47 + 680q46 + 885q45 + 1125q44 + 1399q43 + 1705q42 + 2040q41 + 2399q40 + 2775q39 + 3160q38 + 3546q37 + 3923q36 + 4281q35 + 4610q34 + 4901q33 + 5145q32 + A.1. THE CASE J W0 = J 129

5335q31 + 5465q30 + 5531q29 + 5531q28 + 5465q27 + 5335q26 + 5145q25 + 4901q24 + 4610q23 + 4281q22 + 3923q21 + 3546q20 + 3160q19 + 2775q18 + 2399q17 + 2040q16 + 1705q15 + 1399q14 + 1125q13 + 885q12 + 680q11 + 509q10 + 370q9 + 260q8 + 176q7 + 114q6 + 70q5 + 40q4 + 21q3 + 10q2 + 4q + 1 k = q57

– J = {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 4, 5, 6}, {3, 4, 5, 6}, {4, 5, 6, 7}; type of P : A4 v = q53 + 3q52 + 6q51 + 11q50 + 19q49 + 31q48 + 47q47 + 68q46 + 95q45 + 129q44 + 170q43 + 218q42 + 273q41 + 335q40 + 403q39 + 476q38 + 553q37 + 632q36 + 711q35 + 788q34 +862q33 +930q32 +990q31 +1040q30 +1079q29 +1106q28 +1120q27 +1120q26 + 1106q25 +1079q24 +1040q23 +990q22 +930q21 +862q20 +788q19 +711q18 +632q17 + 553q16 + 476q15 + 403q14 + 335q13 + 273q12 + 218q11 + 170q10 + 129q9 + 95q8 + 68q7 + 47q6 + 31q5 + 19q4 + 11q3 + 6q2 + 3q + 1 k = q53 – J = {1, 2, 3, 5}, {1, 2, 3, 6}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 4, 7}, {1, 2, 5, 6}, {1, 2, 6, 7}, {1, 3, 5, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}, {2, 3, 5, 6}, {2, 3, 6, 7}; type of P : A2A1A1 v = q58 + 3q57 + 8q56 + 17q55 + 33q54 + 58q53 + 96q52 + 150q51 + 224q50 + 322q49 + 447q48 + 603q47 + 791q46 + 1014q45 + 1270q44 + 1560q43 + 1879q42 + 2225q41 + 2590q40 + 2969q39 + 3352q38 + 3731q37 + 4096q36 + 4437q35 + 4745q34 + 5010q33 + 5226q32 + 5384q31 + 5482q30 + 5514q29 + 5482q28 + 5384q27 + 5226q26 + 5010q25 + 4745q24 + 4437q23 + 4096q22 + 3731q21 + 3352q20 + 2969q19 + 2590q18 + 2225q17 + 1879q16 +1560q15 +1270q14 +1014q13 +791q12 +603q11 +447q10 +322q9 +224q8 + 150q7 + 96q6 + 58q5 + 33q4 + 17q3 + 8q2 + 3q + 1 k = q58 – J = {1, 2, 4, 5}, {1, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}, {2, 3, 4, 6}, {2, 3, 4, 7}, {2, 4, 5, 7}, {2, 5, 6, 7}, {3, 4, 5, 7}, {3, 5, 6, 7}; type of P : A3A1 v = q56 + 3q55 + 7q54 + 14q53 + 26q52 + 44q51 + 70q50 + 106q49 + 154q48 + 216q47 + 293q46 +387q45 +498q44 +627q43 +772q42 +933q41 +1107q40 +1292q39 +1483q38 + 1677q37 + 1869q36 + 2054q35 + 2227q34 + 2383q33 + 2518q32 + 2627q31 + 2708q30 + 2757q29 + 2774q28 + 2757q27 + 2708q26 + 2627q25 + 2518q24 + 2383q23 + 2227q22 + 2054q21+1869q20+1677q19+1483q18+1292q17+1107q16+933q15+772q14+627q13+ 498q12+387q11+293q10+216q9+154q8+106q7+70q6+44q5+26q4+14q3+7q2+3q+1 k = q56

– J = {1, 2, 5, 7}, {2, 3, 5, 7}; type of P : A1A1A1A1 v = q59 + 3q58 + 9q57 + 19q56 + 39q55 + 69q54 + 118q53 + 186q52 + 284q51 + 412q50 + 581q49 + 791q48 + 1050q47 + 1358q46 + 1717q45 + 2127q44 + 2582q43 + 3082q42 + 3612q41 + 4172q40 + 4739q39 + 5313q38 + 5866q37 + 6398q36 + 6880q35 + 7312q34 + 7669q33 + 7951q32 + 8141q31 + 8239q30 + 8239q29 + 8141q28 + 7951q27 + 7669q26 + 7312q25 + 6880q24 + 6398q23 + 5866q22 + 5313q21 + 4739q20 + 4172q19 + 3612q18 + 3082q17 + 2582q16 + 2127q15 + 1717q14 + 1358q13 + 1050q12 + 791q11 + 581q10 + 412q9 + 284q8 + 186q7 + 118q6 + 69q5 + 39q4 + 19q3 + 9q2 + 3q + 1 k = q59

– J = {1, 3, 5, 6}, {1, 3, 6, 7}, {2, 4, 6, 7}, {3, 4, 6, 7}; type of P : A2A2 v = q57 + 3q56 + 7q55 + 15q54 + 28q53 + 48q52 + 78q51 + 120q50 + 176q49 + 250q48 + 343q47 +457q46 +594q45 +754q44 +936q43 +1140q42 +1363q41 +1601q40 +1851q39 + 2107q38 + 2363q37 + 2613q36 + 2851q35 + 3069q34 + 3262q33 + 3424q32 + 3550q31 + 3636q30 + 3680q29 + 3680q28 + 3636q27 + 3550q26 + 3424q25 + 3262q24 + 3069q23 + 2851q22 + 2613q21 + 2363q20 + 2107q19 + 1851q18 + 1601q17 + 1363q16 + 1140q15 + 936q14 + 754q13 + 594q12 + 457q11 + 343q10 + 250q9 + 176q8 + 120q7 + 78q6 + 48q5 + 28q4 + 15q3 + 7q2 + 3q + 1 k = q57

– J = {2, 3, 4, 5}; type of P : D4 v = q51 + 3q50 + 6q49 + 10q48 + 17q47 + 27q46 + 40q45 + 56q44 + 77q43 + 103q42 + 133q41 + 167q40 + 206q39 + 250q38 + 296q37 + 344q36 + 394q35 + 446q34 + 495q33 + 541q32 + 584q31 + 624q30 + 656q29 + 680q28 + 697q27 + 707q26 + 707q25 + 697q24 + 130APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

680q23 + 656q22 + 624q21 + 584q20 + 541q19 + 495q18 + 446q17 + 394q16 + 344q15 + 296q14 + 250q13 + 206q12 + 167q11 + 133q10 + 103q9 + 77q8 + 56q7 + 40q6 + 27q5 + 17q4 + 10q3 + 6q2 + 3q + 1 k = q51

– J = {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}; type of P : D5 v = q43 + 2q42 + 3q41 + 4q40 + 6q39 + 9q38 + 12q37 + 15q36 + 19q35 + 24q34 + 29q33 + 34q32 + 39q31 + 45q30 + 50q29 + 55q28 + 60q27 + 65q26 + 68q25 + 70q24 + 72q23 + 74q22 + 74q21 + 72q20 + 70q19 + 68q18 + 65q17 + 60q16 + 55q15 + 50q14 + 45q13 + 39q12 + 34q11 + 29q10 + 24q9 + 19q8 + 15q7 + 12q6 + 9q5 + 6q4 + 4q3 + 3q2 + 2q + 1 k = q43 – J = {1, 2, 3, 4, 6}, {1, 2, 3, 4, 7}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 7}, {1, 4, 5, 6, 7}; type of P : A4A1 v = q52 + 2q51 + 4q50 + 7q49 + 12q48 + 19q47 + 28q46 + 40q45 + 55q44 + 74q43 + 96q42 + 122q41 + 151q40 + 184q39 + 219q38 + 257q37 + 296q36 + 336q35 + 375q34 + 413q33 + 449q32 + 481q31 + 509q30 + 531q29 + 548q28 + 558q27 + 562q26 + 558q25 + 548q24 + 531q23 + 509q22 + 481q21 + 449q20 + 413q19 + 375q18 + 336q17 + 296q16 + 257q15 + 219q14 + 184q13 + 151q12 + 122q11 + 96q10 + 74q9 + 55q8 + 40q7 + 28q6 + 19q5 + 12q4 + 7q3 + 4q2 + 2q + 1 k = q52

– J = {1, 2, 3, 5, 6}, {1, 2, 3, 6, 7}, {1, 2, 4, 6, 7}; type of P : A2A2A1 v = q56 + 2q55 + 5q54 + 10q53 + 18q52 + 30q51 + 48q50 + 72q49 + 104q48 + 146q47 + 197q46 + 260q45 + 334q44 + 420q43 + 516q42 + 624q41 + 739q40 + 862q39 + 989q38 + 1118q37 + 1245q36 + 1368q35 + 1483q34 + 1586q33 + 1676q32 + 1748q31 + 1802q30 + 1834q29 + 1846q28 + 1834q27 + 1802q26 + 1748q25 + 1676q24 + 1586q23 + 1483q22 + 1368q21 +1245q20 +1118q19 +989q18 +862q17 +739q16 +624q15 +516q14 +420q13 + 334q12+260q11+197q10+146q9+104q8+72q7+48q6+30q5+18q4+10q3+5q2+2q+1 k = q56

– J = {1, 2, 3, 5, 7}; type of P : A2A1A1A1 v = q57 + 2q56 + 6q55 + 11q54 + 22q53 + 36q52 + 60q51 + 90q50 + 134q49 + 188q48 + 259q47 +344q46 +447q45 +567q44 +703q43 +857q42 +1022q41 +1203q40 +1387q39 + 1582q38 + 1770q37 + 1961q36 + 2135q35 + 2302q34 + 2443q33 + 2567q32 + 2659q31 + 2725q30 + 2757q29 + 2757q28 + 2725q27 + 2659q26 + 2567q25 + 2443q24 + 2302q23 + 2135q22 + 1961q21 + 1770q20 + 1582q19 + 1387q18 + 1203q17 + 1022q16 + 857q15 + 703q14 + 567q13 + 447q12 + 344q11 + 259q10 + 188q9 + 134q8 + 90q7 + 60q6 + 36q5 + 22q4 + 11q3 + 6q2 + 2q + 1 k = q57

– J = {1, 2, 4, 5, 7}, {1, 2, 5, 6, 7}; type of P : A3A1A1 v = q55 + 2q54 + 5q53 + 9q52 + 17q51 + 27q50 + 43q49 + 63q48 + 91q47 + 125q46 + 168q45 + 219q44 + 279q43 + 348q42 + 424q41 + 509q40 + 598q39 + 694q38 + 789q37 + 888q36 + 981q35 + 1073q34 + 1154q33 + 1229q32 + 1289q31 + 1338q30 + 1370q29 + 1387q28 + 1387q27 + 1370q26 + 1338q25 + 1289q24 + 1229q23 + 1154q22 + 1073q21 + 981q20 + 888q19 + 789q18 + 694q17 + 598q16 + 509q15 + 424q14 + 348q13 + 279q12 + 219q11 + 168q10 + 125q9 + 91q8 + 63q7 + 43q6 + 27q5 + 17q4 + 9q3 + 5q2 + 2q + 1 k = q55

– J = {1, 3, 4, 5, 6}, {2, 4, 5, 6, 7}, {3, 4, 5, 6, 7}; type of P : A5 v = q48 +2q47 +3q46 +5q45 +8q44 +12q43 +17q42 +23q41 +30q40 +39q39 +49q38 + 60q37 +72q36 +85q35 +98q34 +112q33 +126q32 +139q31 +151q30 +162q29 +172q28 + 180q27 + 186q26 + 189q25 + 190q24 + 189q23 + 186q22 + 180q21 + 172q20 + 162q19 + 151q18 + 139q17 + 126q16 + 112q15 + 98q14 + 85q13 + 72q12 + 60q11 + 49q10 + 39q9 + 30q8 + 23q7 + 17q6 + 12q5 + 8q4 + 5q3 + 3q2 + 2q + 1 k = q48

– J = {1, 3, 4, 6, 7}, {1, 3, 5, 6, 7}, {2, 3, 4, 6, 7}, {2, 3, 5, 6, 7}; type of P : A3A2 v = q54 + 2q53 + 4q52 + 8q51 + 14q50 + 22q49 + 34q48 + 50q47 + 70q46 + 96q45 + 127q44 + 164q43 + 207q42 + 256q41 + 309q40 + 368q39 + 430q38 + 494q37 + 559q36 + A.1. THE CASE J W0 = J 131

624q35 + 686q34 + 744q33 + 797q32 + 842q31 + 879q30 + 906q29 + 923q28 + 928q27 + 923q26 + 906q25 + 879q24 + 842q23 + 797q22 + 744q21 + 686q20 + 624q19 + 559q18 + 494q17 + 430q16 + 368q15 + 309q14 + 256q13 + 207q12 + 164q11 + 127q10 + 96q9 + 70q8 + 50q7 + 34q6 + 22q5 + 14q4 + 8q3 + 4q2 + 2q + 1 k = q54

– J = {2, 3, 4, 5, 7}; type of P : D4A1 v = q50 +2q49 +4q48 +6q47 +11q46 +16q45 +24q44 +32q43 +45q42 +58q41 +75q40 + 92q39 + 114q38 + 136q37 + 160q36 + 184q35 + 210q34 + 236q33 + 259q32 + 282q31 + 302q30 + 322q29 + 334q28 + 346q27 + 351q26 + 356q25 + 351q24 + 346q23 + 334q22 + 322q21 + 302q20 + 282q19 + 259q18 + 236q17 + 210q16 + 184q15 + 160q14 + 136q13 + 114q12 + 92q11 + 75q10 + 58q9 + 45q8 + 32q7 + 24q6 + 16q5 + 11q4 + 6q3 + 4q2 + 2q + 1 k = q50

– J = {1, 2, 3, 4, 5, 6}; type of P : E6 v = q27 +q26 +q25 +q24 +q23 +2q22 +2q21 +2q20 +2q19 +3q18 +3q17 +3q16 +3q15 + 3q14 + 3q13 + 3q12 + 3q11 + 3q10 + 3q9 + 2q8 + 2q7 + 2q6 + 2q5 + q4 + q3 + q2 + q + 1 k = q27

– J = {1, 2, 3, 4, 5, 7}; type of P : D5A1 v = q42 + q41 + 2q40 + 2q39 + 4q38 + 5q37 + 7q36 + 8q35 + 11q34 + 13q33 + 16q32 + 18q31 + 21q30 + 24q29 + 26q28 + 29q27 + 31q26 + 34q25 + 34q24 + 36q23 + 36q22 + 38q21 + 36q20 + 36q19 + 34q18 + 34q17 + 31q16 + 29q15 + 26q14 + 24q13 + 21q12 + 18q11 + 16q10 + 13q9 + 11q8 + 8q7 + 7q6 + 5q5 + 4q4 + 2q3 + 2q2 + q + 1 k = q42

– J = {1, 2, 3, 4, 6, 7}; type of P : A4A2 v = q50 + q49 + 2q48 + 4q47 + 6q46 + 9q45 + 13q44 + 18q43 + 24q42 + 32q41 + 40q40 + 50q39 +61q38 +73q37 +85q36 +99q35 +112q34 +125q33 +138q32 +150q31 +161q30 + 170q29 + 178q28 + 183q27 + 187q26 + 188q25 + 187q24 + 183q23 + 178q22 + 170q21 + 161q20 +150q19 +138q18 +125q17 +112q16 +99q15 +85q14 +73q13 +61q12 +50q11 + 40q10 + 32q9 + 24q8 + 18q7 + 13q6 + 9q5 + 6q4 + 4q3 + 2q2 + q + 1 k = q50

– J = {1, 2, 3, 5, 6, 7}; type of P : A3A2A1 v = q53 + q52 + 3q51 + 5q50 + 9q49 + 13q48 + 21q47 + 29q46 + 41q45 + 55q44 + 72q43 + 92q42 + 115q41 + 141q40 + 168q39 + 200q38 + 230q37 + 264q36 + 295q35 + 329q34 + 357q33 + 387q32 + 410q31 + 432q30 + 447q29 + 459q28 + 464q27 + 464q26 + 459q25 + 447q24 + 432q23 + 410q22 + 387q21 + 357q20 + 329q19 + 295q18 + 264q17 + 230q16 + 200q15 + 168q14 + 141q13 + 115q12 + 92q11 + 72q10 + 55q9 + 41q8 + 29q7 + 21q6 + 13q5 + 9q4 + 5q3 + 3q2 + q + 1 k = q53

– J = {1, 2, 4, 5, 6, 7}; type of P : A5A1 v = q47+q46+2q45+3q44+5q43+7q42+10q41+13q40+17q39+22q38+27q37+33q36+ 39q35 +46q34 +52q33 +60q32 +66q31 +73q30 +78q29 +84q28 +88q27 +92q26 +94q25 + 95q24 +95q23 +94q22 +92q21 +88q20 +84q19 +78q18 +73q17 +66q16 +60q15 +52q14 + 46q13 +39q12 +33q11 +27q10 +22q9 +17q8 +13q7 +10q6 +7q5 +5q4 +3q3 +2q2 +q+1 k = q47

– J = {1, 3, 4, 5, 6, 7}; type of P : A6 v = q42 + q41 + q40 + 2q39 + 3q38 + 4q37 + 5q36 + 7q35 + 8q34 + 10q33 + 12q32 + 14q31 + 16q30 + 18q29 + 20q28 + 22q27 + 24q26 + 25q25 + 26q24 + 27q23 + 28q22 + 28q21 + 28q20 + 27q19 + 26q18 + 25q17 + 24q16 + 22q15 + 20q14 + 18q13 + 16q12 + 14q11 + 12q10 + 10q9 + 8q8 + 7q7 + 5q6 + 4q5 + 3q4 + 2q3 + q2 + q + 1 k = q42

– J = {2, 3, 4, 5, 6, 7}; type of P : D6 v = q33 + q32 + q31 + q30 + 2q29 + 2q28 + 3q27 + 3q26 + 4q25 + 4q24 + 5q23 + 5q22 + 6q21 + 6q20 + 6q19 + 6q18 + 7q17 + 7q16 + 6q15 + 6q14 + 6q13 + 6q12 + 5q11 + 5q10 + 4q9 + 4q8 + 3q7 + 3q6 + 2q5 + 2q4 + q3 + q2 + q + 1 k = q33 132APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

E8 – J = {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}; type of P : A1 v = q119 + 7q118 + 28q117 + 84q116 + 210q115 + 462q114 + 924q113 + 1716q112 + 3002q111 + 4998q110 + 7980q109 + 12292q108 + 18353q107 + 26663q106 + 37807q105 + 52457q104+71372q103+95396q102+125453q101+162539q100+207711q99+262073q98+ 326760q97+402920q96+491693q95+594187q94+711453q93+844459q92+994064q91+ 1160992q90 + 1345806q89 + 1548882q88 + 1770386q87 + 2010254q86 + 2268175q85 + 2543577q84 + 2835617q83 + 3143175q82 + 3464854q81 + 3798986q80 + 4143642q79 + 4496646q78 + 4855594q77 + 5217878q76 + 5580715q75 + 5941181q74 + 6296247q73 + 6642817q72 + 6977769q71 + 7297999q70 + 7600465q69 + 7882231q68 + 8140509q67 + 8372699q66 + 8576428q65 + 8749588q64 + 8890369q63 + 8997287q62 + 9069207q61 + 9105361q60 + 9105361q59 + 9069207q58 + 8997287q57 + 8890369q56 + 8749588q55 + 8576428q54 + 8372699q53 + 8140509q52 + 7882231q51 + 7600465q50 + 7297999q49 + 6977769q48 + 6642817q47 + 6296247q46 + 5941181q45 + 5580715q44 + 5217878q43 + 4855594q42 + 4496646q41 + 4143642q40 + 3798986q39 + 3464854q38 + 3143175q37 + 2835617q36 + 2543577q35 + 2268175q34 + 2010254q33 + 1770386q32 + 1548882q31 + 1345806q30+1160992q29+994064q28+844459q27+711453q26+594187q25+491693q24+ 402920q23 +326760q22 +262073q21 +207711q20 +162539q19 +125453q18 +95396q17 + 71372q16+52457q15+37807q14+26663q13+18353q12+12292q11+7980q10+4998q9+ 3002q8 + 1716q7 + 924q6 + 462q5 + 210q4 + 84q3 + 28q2 + 7q + 1 k = q119 – J = {1, 2}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {2, 3}, {2, 5}, {2, 6}, {2, 7}, {2, 8}, {3, 5}, {3, 6}, {3, 7}, {3, 8}, {4, 6}, {4, 7}, {4, 8}, {5, 7}, {5, 8}, {6, 8}; type of P : A1A1 v = q118 + 6q117 + 22q116 + 62q115 + 148q114 + 314q113 + 610q112 + 1106q111 + 1896q110 + 3102q109 + 4878q108 + 7414q107 + 10939q106 + 15724q105 + 22083q104 + 30374q103 +40998q102 +54398q101 +71055q100 +91484q99 +116227q98 +145846q97 + 180914q96+222006q95+269687q94+324500q93+386953q92+457506q91+536558q90+ 624434q89+721372q88+827510q87+942876q86+1067378q85+1200797q84+1342780q83+ 1492837q82 + 1650338q81 + 1814516q80 + 1984470q79 + 2159172q78 + 2337474q77 + 2518120q76 + 2699758q75 + 2880957q74 + 3060224q73 + 3236023q72 + 3406794q71 + 3570975q70 + 3727024q69 + 3873441q68 + 4008790q67 + 4131719q66 + 4240980q65 + 4335448q64 + 4414140q63 + 4476229q62 + 4521058q61 + 4548149q60 + 4557212q59 + 4548149q58 + 4521058q57 + 4476229q56 + 4414140q55 + 4335448q54 + 4240980q53 + 4131719q52 + 4008790q51 + 3873441q50 + 3727024q49 + 3570975q48 + 3406794q47 + 3236023q46 + 3060224q45 + 2880957q44 + 2699758q43 + 2518120q42 + 2337474q41 + 2159172q40 + 1984470q39 + 1814516q38 + 1650338q37 + 1492837q36 + 1342780q35 + 1200797q34+1067378q33+942876q32+827510q31+721372q30+624434q29+536558q28+ 457506q27+386953q26+324500q25+269687q24+222006q23+180914q22+145846q21+ 116227q20 + 91484q19 + 71055q18 + 54398q17 + 40998q16 + 30374q15 + 22083q14 + 15724q13 + 10939q12 + 7414q11 + 4878q10 + 3102q9 + 1896q8 + 1106q7 + 610q6 + 314q5 + 148q4 + 62q3 + 22q2 + 6q + 1 k = q118

– J = {1, 3}, {2, 4}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8}; type of P : A2 v = q117 + 6q116 + 21q115 + 57q114 + 132q113 + 273q112 + 519q111 + 924q110 + 1559q109 + 2515q108 + 3906q107 + 5871q106 + 8576q105 + 12216q104 + 17015q103 + 23226q102 + 31131q101 + 41039q100 + 53283q99 + 68217q98 + 86211q97 + 107645q96 + 132904q95+162371q94+196418q93+235398q92+279637q91+329424q90+385003q89+ 446565q88+514238q87+588079q86+668069q85+754106q84+846000q83+943471q82+ 1046146q81 + 1153558q80 + 1265150q79 + 1380278q78 + 1498214q77 + 1618154q76 + 1739226q75 + 1860498q74 + 1980991q73 + 2099692q72 + 2215564q71 + 2327561q70 + 2434644q69 + 2535794q68 + 2630027q67 + 2716410q66 + 2794072q65 + 2862217q64 + 2920139q63 + 2967232q62 + 3002998q61 + 3027057q60 + 3039152q59 + 3039152q58 + 3027057q57 + 3002998q56 + 2967232q55 + 2920139q54 + 2862217q53 + 2794072q52 + 2716410q51 + 2630027q50 + 2535794q49 + 2434644q48 + 2327561q47 + 2215564q46 + 2099692q45 + 1980991q44 + 1860498q43 + 1739226q42 + 1618154q41 + 1498214q40 + 1380278q39 + 1265150q38 + 1153558q37 + 1046146q36 + 943471q35 + 846000q34 + 754106q33+668069q32+588079q31+514238q30+446565q29+385003q28+329424q27+ A.1. THE CASE J W0 = J 133

279637q26 +235398q25 +196418q24 +162371q23 +132904q22 +107645q21 +86211q20 + 68217q19 + 53283q18 + 41039q17 + 31131q16 + 23226q15 + 17015q14 + 12216q13 + 8576q12 + 5871q11 + 3906q10 + 2515q9 + 1559q8 + 924q7 + 519q6 + 273q5 + 132q4 + 57q3 + 21q2 + 6q + 1 k = q117

– J = {1, 2, 3},... ; type of P : A2A1 v = q116 +5q115 +16q114 +41q113 +91q112 +182q111 +337q110 +587q109 +972q108 + 1543q107 + 2363q106 + 3508q105 + 5068q104 + 7148q103 + 9867q102 + 13359q101 + 17772q100 + 23267q99 + 30016q98 + 38201q97 + 48010q96 + 59635q95 + 73269q94 + 89102q93 +107316q92 +128082q91 +151555q90 +177869q89 +207134q88 +239431q87 + 274807q86+313272q85+354797q84+399309q83+446691q82+496780q81+549366q80+ 604192q79+660958q78+719320q77+778894q76+839260q75+899966q74+960532q73+ 1020459q72 + 1079233q71 + 1136331q70 + 1191230q69 + 1243414q68 + 1292380q67 + 1337647q66 + 1378763q65 + 1415309q64 + 1446908q63 + 1473231q62 + 1494001q61 + 1508997q60 + 1518060q59 + 1521092q58 + 1518060q57 + 1508997q56 + 1494001q55 + 1473231q54 + 1446908q53 + 1415309q52 + 1378763q51 + 1337647q50 + 1292380q49 + 1243414q48 + 1191230q47 + 1136331q46 + 1079233q45 + 1020459q44 + 960532q43 + 899966q42+839260q41+778894q40+719320q39+660958q38+604192q37+549366q36+ 496780q35+446691q34+399309q33+354797q32+313272q31+274807q30+239431q29+ 207134q28 +177869q27 +151555q26 +128082q25 +107316q24 +89102q23 +73269q22 + 59635q21 + 48010q20 + 38201q19 + 30016q18 + 23267q17 + 17772q16 + 13359q15 + 9867q14 +7148q13 +5068q12 +3508q11 +2363q10 +1543q9 +972q8 +587q7 +337q6 + 182q5 + 91q4 + 41q3 + 16q2 + 5q + 1 k = q116

– J = {1, 2, 5},... ; type of P : A1A1A1 v = q117+5q116+17q115+45q114+103q113+211q112+399q111+707q110+1189q109+ 1913q108 + 2965q107 + 4449q106 + 6490q105 + 9234q104 + 12849q103 + 17525q102 + 23473q101 + 30925q100 + 40130q99 + 51354q98 + 64873q97 + 80973q96 + 99941q95 + 122065q94+147622q93+176878q92+210075q91+247431q90+289127q89+335307q88+ 386065q87+441445q86+501431q85+565947q84+634850q83+707930q82+784907q81+ 865431q80 + 949085q79 + 1035385q78 + 1123787q77 + 1213687q76 + 1304433q75 + 1395325q74 + 1485632q73 + 1574592q72 + 1661431q71 + 1745363q70 + 1825612q69 + 1901412q68 + 1972029q67 + 2036761q66 + 2094958q65 + 2146022q64 + 2189426q63 + 2224714q62 + 2251515q61 + 2269543q60 + 2278606q59 + 2278606q58 + 2269543q57 + 2251515q56 + 2224714q55 + 2189426q54 + 2146022q53 + 2094958q52 + 2036761q51 + 1972029q50 + 1901412q49 + 1825612q48 + 1745363q47 + 1661431q46 + 1574592q45 + 1485632q44 + 1395325q43 + 1304433q42 + 1213687q41 + 1123787q40 + 1035385q39 + 949085q38+865431q37+784907q36+707930q35+634850q34+565947q33+501431q32+ 441445q31+386065q30+335307q29+289127q28+247431q27+210075q26+176878q25+ 147622q24 + 122065q23 + 99941q22 + 80973q21 + 64873q20 + 51354q19 + 40130q18 + 30925q17 +23473q16 +17525q15 +12849q14 +9234q13 +6490q12 +4449q11 +2965q10 + 1913q9 + 1189q8 + 707q7 + 399q6 + 211q5 + 103q4 + 45q3 + 17q2 + 5q + 1 k = q117

– J = {1, 3, 4},... ; type of P : A3 v = q114 + 5q113 + 15q112 + 36q111 + 76q110 + 146q109 + 261q108 + 441q107 + 711q106+1102q105+1652q104+2406q103+3416q102+4742q101+6451q100+8617q99+ 11321q98 + 14650q97 + 18695q96 + 23551q95 + 29315q94 + 36084q93 + 43954q92 + 53018q91 + 63362q90 + 75064q89 + 88193q88 + 102805q87 + 118941q86 + 136626q85 + 155866q84+176646q83+198931q82+222663q81+247760q80+274117q79+301606q78+ 330075q77+359352q76+389245q75+419542q74+450015q73+480424q72+510517q71+ 540035q70+568716q69+596296q68+622514q67+647118q66+669866q65+690529q64+ 708897q63+724780q62+738011q61+748451q60+755990q59+760546q58+762070q57+ 760546q56+755990q55+748451q54+738011q53+724780q52+708897q51+690529q50+ 669866q49+647118q48+622514q47+596296q46+568716q45+540035q44+510517q43+ 480424q42+450015q41+419542q40+389245q39+359352q38+330075q37+301606q36+ 274117q35+247760q34+222663q33+198931q32+176646q31+155866q30+136626q29+ 134APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

118941q28 + 102805q27 + 88193q26 + 75064q25 + 63362q24 + 53018q23 + 43954q22 + 36084q21 + 29315q20 + 23551q19 + 18695q18 + 14650q17 + 11321q16 + 8617q15 + 6451q14 +4742q13 +3416q12 +2406q11 +1652q10 +1102q9 +711q8 +441q7 +261q6 + 146q5 + 76q4 + 36q3 + 15q2 + 5q + 1 k = q114

– J = {1, 2, 3, 4},... ; type of P : A4 v = q110 +4q109 +10q108 +21q107 +40q106 +71q105 +119q104 +190q103 +291q102 + 431q101 + 621q100 + 873q99 + 1200q98 + 1617q97 + 2140q96 + 2787q95 + 3577q94 + 4529q93 +5662q92 +6996q91 +8551q90 +10346q89 +12399q88 +14726q87 +17340q86 + 20253q85 + 23475q84 + 27011q83 + 30862q82 + 35025q81 + 39493q80 + 44255q79 + 49296q78 + 54594q77 + 60122q76 + 65850q75 + 71744q74 + 77765q73 + 83871q72 + 90015q71 +96147q70 +102217q69 +108174q68 +113964q67 +119533q66 +124828q65 + 129797q64+134392q63+138568q62+142281q61+145491q60+148165q59+150275q58+ 151799q57+152721q56+153030q55+152721q54+151799q53+150275q52+148165q51+ 145491q50+142281q49+138568q48+134392q47+129797q46+124828q45+119533q44+ 113964q43 + 108174q42 + 102217q41 + 96147q40 + 90015q39 + 83871q38 + 77765q37 + 71744q36 + 65850q35 + 60122q34 + 54594q33 + 49296q32 + 44255q31 + 39493q30 + 35025q29 + 30862q28 + 27011q27 + 23475q26 + 20253q25 + 17340q24 + 14726q23 + 12399q22 + 10346q21 + 8551q20 + 6996q19 + 5662q18 + 4529q17 + 3577q16 + 2787q15 + 2140q14 + 1617q13 + 1200q12 + 873q11 + 621q10 + 431q9 + 291q8 + 190q7 + 119q6 + 71q5 + 40q4 + 21q3 + 10q2 + 4q + 1 k = q110

– J = {1, 2, 3, 5},... ; type of P : A2A1A1 v = q115 + 4q114 + 12q113 + 29q112 + 62q111 + 120q110 + 217q109 + 370q108 + 602q107+941q106+1422q105+2086q104+2982q103+4166q102+5701q101+7658q100+ 10114q99 + 13153q98 + 16863q97 + 21338q96 + 26672q95 + 32963q94 + 40306q93 + 48796q92 + 58520q91 + 69562q90 + 81993q89 + 95876q88 + 111258q87 + 128173q86 + 146634q85+166638q84+188159q83+211150q82+235541q81+261239q80+288127q79+ 316065q78+344893q77+374427q76+404467q75+434793q74+465173q73+495359q72+ 525100q71+554133q70+582198q69+609032q68+634382q67+657998q66+679649q65+ 699114q64+716195q63+730713q62+742518q61+751483q60+757514q59+760546q58+ 760546q57+757514q56+751483q55+742518q54+730713q53+716195q52+699114q51+ 679649q50+657998q49+634382q48+609032q47+582198q46+554133q45+525100q44+ 495359q43+465173q42+434793q41+404467q40+374427q39+344893q38+316065q37+ 288127q36+261239q35+235541q34+211150q33+188159q32+166638q31+146634q30+ 128173q29 + 111258q28 + 95876q27 + 81993q26 + 69562q25 + 58520q24 + 48796q23 + 40306q22 + 32963q21 + 26672q20 + 21338q19 + 16863q18 + 13153q17 + 10114q16 + 7658q15 +5701q14 +4166q13 +2982q12 +2086q11 +1422q10 +941q9 +602q8 +370q7 + 217q6 + 120q5 + 62q4 + 29q3 + 12q2 + 4q + 1 k = q115

– J = {1, 2, 4, 5},... ; type of P : A3A1 v = q113 +4q112 +11q111 +25q110 +51q109 +95q108 +166q107 +275q106 +436q105 + 666q104 +986q103 +1420q102 +1996q101 +2746q100 +3705q99 +4912q98 +6409q97 + 8241q96 + 10454q95 + 13097q94 + 16218q93 + 19866q92 + 24088q91 + 28930q90 + 34432q89 + 40632q88 + 47561q87 + 55244q86 + 63697q85 + 72929q84 + 82937q83 + 93709q82 +105222q81 +117441q80 +130319q79 +143798q78 +157808q77 +172267q76 + 187085q75+202160q74+217382q73+232633q72+247791q71+262726q70+277309q69+ 291407q68+304889q67+317625q66+329493q65+340373q64+350156q63+358741q62+ 366039q61+371972q60+376479q59+379511q58+381035q57+381035q56+379511q55+ 376479q54+371972q53+366039q52+358741q51+350156q50+340373q49+329493q48+ 317625q47+304889q46+291407q45+277309q44+262726q43+247791q42+232633q41+ 217382q40+202160q39+187085q38+172267q37+157808q36+143798q35+130319q34+ 117441q33 + 105222q32 + 93709q31 + 82937q30 + 72929q29 + 63697q28 + 55244q27 + 47561q26 + 40632q25 + 34432q24 + 28930q23 + 24088q22 + 19866q21 + 16218q20 + 13097q19 + 10454q18 + 8241q17 + 6409q16 + 4912q15 + 3705q14 + 2746q13 + 1996q12 + A.1. THE CASE J W0 = J 135

1420q11 +986q10 +666q9 +436q8 +275q7 +166q6 +95q5 +51q4 +25q3 +11q2 +4q +1 k = q113

– J = {1, 2, 5, 7},... ; type of P : A1A1A1A1 v = q116 + 4q115 + 13q114 + 32q113 + 71q112 + 140q111 + 259q110 + 448q109 + 741q108 + 1172q107 + 1793q106 + 2656q105 + 3834q104 + 5400q103 + 7449q102 + 10076q101 + 13397q100 + 17528q99 + 22602q98 + 28752q97 + 36121q96 + 44852q95 + 55089q94 + 66976q93 + 80646q92 + 96232q91 + 113843q90 + 133588q89 + 155539q88 + 179768q87+206297q86+235148q85+266283q84+299664q83+335186q82+372744q81+ 412163q80+453268q79+495817q78+539568q77+584219q76+629468q75+674965q74+ 720360q73+765272q72+809320q71+852111q70+893252q69+932360q68+969052q67+ 1002977q66 + 1033784q65 + 1061174q64 + 1084848q63 + 1104578q62 + 1120136q61 + 1131379q60 + 1138164q59 + 1140442q58 + 1138164q57 + 1131379q56 + 1120136q55 + 1104578q54 + 1084848q53 + 1061174q52 + 1033784q51 + 1002977q50 + 969052q49 + 932360q48+893252q47+852111q46+809320q45+765272q44+720360q43+674965q42+ 629468q41+584219q40+539568q39+495817q38+453268q37+412163q36+372744q35+ 335186q34+299664q33+266283q32+235148q31+206297q30+179768q29+155539q28+ 133588q27 + 113843q26 + 96232q25 + 80646q24 + 66976q23 + 55089q22 + 44852q21 + 36121q20 + 28752q19 + 22602q18 + 17528q17 + 13397q16 + 10076q15 + 7449q14 + 5400q13 + 3834q12 + 2656q11 + 1793q10 + 1172q9 + 741q8 + 448q7 + 259q6 + 140q5 + 71q4 + 32q3 + 13q2 + 4q + 1 k = q116

– J = {1, 3, 5, 6},... ; type of P : A2A2 v = q114 +4q113 +11q112 +26q111 +54q110 +102q109 +181q108 +304q107 +487q106 + 752q105 +1124q104 +1632q103 +2312q102 +3204q101 +4351q100 +5804q99 +7617q98 + 9846q97 + 12553q96 + 15802q95 + 19655q94 + 24178q93 + 29436q92 + 35488q91 + 42392q90 + 50202q89 + 58961q88 + 68706q87 + 79467q86 + 91258q85 + 104082q84 + 117932q83+132783q82+148594q81+165314q80+182872q79+201180q78+220140q77+ 239638q76+259542q75+279714q74+300004q73+320248q72+340280q71+359931q70+ 379022q69+397378q68+414830q67+431206q66+446344q65+460097q64+472322q63+ 482890q62+491696q61+498645q60+503660q59+506692q58+507708q57+506692q56+ 503660q55+498645q54+491696q53+482890q52+472322q51+460097q50+446344q49+ 431206q48+414830q47+397378q46+379022q45+359931q44+340280q43+320248q42+ 300004q41+279714q40+259542q39+239638q38+220140q37+201180q36+182872q35+ 165314q34 +148594q33 +132783q32 +117932q31 +104082q30 +91258q29 +79467q28 + 68706q27 + 58961q26 + 50202q25 + 42392q24 + 35488q23 + 29436q22 + 24178q21 + 19655q20 +15802q19 +12553q18 +9846q17 +7617q16 +5804q15 +4351q14 +3204q13 + 2312q12 +1632q11 +1124q10 +752q9 +487q8 +304q7 +181q6 +102q5 +54q4 +26q3 + 11q2 + 4q + 1 k = q114

– J = {2, 3, 4, 5}; type of P : D4 v = q108 +4q107 +10q106 +20q105 +37q104 +64q103 +105q102 +164q101 +247q100 + 360q99 +511q98 +708q97 +961q96 +1280q95 +1676q94 +2160q93 +2745q92 +3444q91 + 4269q90 +5232q89 +6345q88 +7620q87 +9067q86 +10696q85 +12513q84 +14524q83 + 16732q82 + 19140q81 + 21745q80 + 24544q79 + 27528q78 + 30688q77 + 34010q76 + 37480q75 + 41077q74 + 44780q73 + 48563q72 + 52400q71 + 56262q70 + 60120q69 + 63940q68 + 67688q67 + 71331q66 + 74836q65 + 78170q64 + 81300q63 + 84195q62 + 86824q61 + 89162q60 + 91184q59 + 92870q58 + 94200q57 + 95161q56 + 95740q55 + 95934q54 + 95740q53 + 95161q52 + 94200q51 + 92870q50 + 91184q49 + 89162q48 + 86824q47 + 84195q46 + 81300q45 + 78170q44 + 74836q43 + 71331q42 + 67688q41 + 63940q40 + 60120q39 + 56262q38 + 52400q37 + 48563q36 + 44780q35 + 41077q34 + 37480q33 + 34010q32 + 30688q31 + 27528q30 + 24544q29 + 21745q28 + 19140q27 + 16732q26 +14524q25 +12513q24 +10696q23 +9067q22 +7620q21 +6345q20 +5232q19 + 4269q18 + 3444q17 + 2745q16 + 2160q15 + 1676q14 + 1280q13 + 961q12 + 708q11 + 511q10 + 360q9 + 247q8 + 164q7 + 105q6 + 64q5 + 37q4 + 20q3 + 10q2 + 4q + 1 k = q108 136APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

– J = {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}; type of P : D5 v = q100 + 3q99 + 6q98 + 10q97 + 16q96 + 25q95 + 38q94 + 55q93 + 77q92 + 105q91 + 141q90 + 186q89 + 241q88 + 307q87 + 385q86 + 477q85 + 585q84 + 710q83 + 852q82 + 1012q81 + 1191q80 + 1391q79 + 1612q78 + 1854q77 + 2115q76 + 2396q75 + 2697q74 + 3018q73 + 3356q72 + 3709q71 + 4074q70 + 4451q69 + 4838q68 + 5232q67 + 5628q66 + 6023q65 + 6414q64 + 6799q63 + 7175q62 + 7537q61 + 7880q60 + 8201q59 + 8499q58 + 8771q57 + 9014q56 + 9223q55 + 9396q54 + 9532q53 + 9632q52 + 9693q51 + 9714q50 + 9693q49 + 9632q48 + 9532q47 + 9396q46 + 9223q45 + 9014q44 + 8771q43 + 8499q42 + 8201q41 + 7880q40 + 7537q39 + 7175q38 + 6799q37 + 6414q36 + 6023q35 + 5628q34 + 5232q33 + 4838q32 + 4451q31 + 4074q30 + 3709q29 + 3356q28 + 3018q27 + 2697q26 + 2396q25 + 2115q24 + 1854q23 + 1612q22 + 1391q21 + 1191q20 + 1012q19 + 852q18 + 710q17 + 585q16 + 477q15 + 385q14 + 307q13 + 241q12 + 186q11 + 141q10 + 105q9 + 77q8 + 55q7 + 38q6 + 25q5 + 16q4 + 10q3 + 6q2 + 3q + 1 k = q100

– J = {1, 2, 3, 4, 6},... ; type of P : A4A1 v = q109 + 3q108 + 7q107 + 14q106 + 26q105 + 45q104 + 74q103 + 116q102 + 175q101 + 256q100 +365q99 +508q98 +692q97 +925q96 +1215q95 +1572q94 +2005q93 +2524q92 + 3138q91 + 3858q90 + 4693q89 + 5653q88 + 6746q87 + 7980q86 + 9360q85 + 10893q84 + 12582q83 + 14429q82 + 16433q81 + 18592q80 + 20901q79 + 23354q78 + 25942q77 + 28652q76 + 31470q75 + 34380q74 + 37364q73 + 40401q72 + 43470q71 + 46545q70 + 49602q69 + 52615q68 + 55559q67 + 58405q66 + 61128q65 + 63700q64 + 66097q63 + 68295q62 + 70273q61 + 72008q60 + 73483q59 + 74682q58 + 75593q57 + 76206q56 + 76515q55 + 76515q54 + 76206q53 + 75593q52 + 74682q51 + 73483q50 + 72008q49 + 70273q48 + 68295q47 + 66097q46 + 63700q45 + 61128q44 + 58405q43 + 55559q42 + 52615q41 + 49602q40 + 46545q39 + 43470q38 + 40401q37 + 37364q36 + 34380q35 + 31470q34 + 28652q33 + 25942q32 + 23354q31 + 20901q30 + 18592q29 + 16433q28 + 14429q27 +12582q26 +10893q25 +9360q24 +7980q23 +6746q22 +5653q21 +4693q20 + 3858q19 + 3138q18 + 2524q17 + 2005q16 + 1572q15 + 1215q14 + 925q13 + 692q12 + 508q11 + 365q10 + 256q9 + 175q8 + 116q7 + 74q6 + 45q5 + 26q4 + 14q3 + 7q2 + 3q + 1 k = q109

– J = {1, 2, 3, 5, 6},... ; type of P : A2A2A1 v = q113 + 3q112 + 8q111 + 18q110 + 36q109 + 66q108 + 115q107 + 189q106 + 298q105 + 454q104 + 670q103 + 962q102 + 1350q101 + 1854q100 + 2497q99 + 3307q98 + 4310q97 + 5536q96+7017q95+8785q94+10870q93+13308q92+16128q91+19360q90+23032q89+ 27170q88 + 31791q87 + 36915q86 + 42552q85 + 48706q84 + 55376q83 + 62556q82 + 70227q81 + 78367q80 + 86947q79 + 95925q78 + 105255q77 + 114885q76 + 124753q75 + 134789q74+144925q73+155079q72+165169q71+175111q70+184820q69+194202q68+ 203176q67+211654q66+219552q65+226792q64+233305q63+239017q62+243873q61+ 247823q60+250822q59+252838q58+253854q57+253854q56+252838q55+250822q54+ 247823q53+243873q52+239017q51+233305q50+226792q49+219552q48+211654q47+ 203176q46+194202q45+184820q44+175111q43+165169q42+155079q41+144925q40+ 134789q39 +124753q38 +114885q37 +105255q36 +95925q35 +86947q34 +78367q33 + 70227q32 + 62556q31 + 55376q30 + 48706q29 + 42552q28 + 36915q27 + 31791q26 + 27170q25 + 23032q24 + 19360q23 + 16128q22 + 13308q21 + 10870q20 + 8785q19 + 7017q18 + 5536q17 + 4310q16 + 3307q15 + 2497q14 + 1854q13 + 1350q12 + 962q11 + 670q10 + 454q9 + 298q8 + 189q7 + 115q6 + 66q5 + 36q4 + 18q3 + 8q2 + 3q + 1 k = q113

– J = {1, 2, 3, 5, 7},... ; type of P : A2A1A1A1 v = q114 + 3q113 + 9q112 + 20q111 + 42q110 + 78q109 + 139q108 + 231q107 + 371q106 + 570q105 +852q104 +1234q103 +1748q102 +2418q101 +3283q100 +4375q99 +5739q98 + 7414q97 + 9449q96 + 11889q95 + 14783q94 + 18180q93 + 22126q92 + 26670q91 + 31850q90 + 37712q89 + 44281q88 + 51595q87 + 59663q86 + 68510q85 + 78124q84 + 88514q83 +99645q82 +111505q81 +124036q80 +137203q79 +150924q78 +165141q77 + 179752q76+194675q75+209792q74+225001q73+240172q72+255187q71+269913q70+ 284220q69+297978q68+311054q67+323328q66+334670q65+344979q64+354135q63+ 362060q62+368653q61+373865q60+377618q59+379896q58+380650q57+379896q56+ A.1. THE CASE J W0 = J 137

377618q55+373865q54+368653q53+362060q52+354135q51+344979q50+334670q49+ 323328q48+311054q47+297978q46+284220q45+269913q44+255187q43+240172q42+ 225001q41+209792q40+194675q39+179752q38+165141q37+150924q36+137203q35+ 124036q34 + 111505q33 + 99645q32 + 88514q31 + 78124q30 + 68510q29 + 59663q28 + 51595q27 + 44281q26 + 37712q25 + 31850q24 + 26670q23 + 22126q22 + 18180q21 + 14783q20 + 11889q19 + 9449q18 + 7414q17 + 5739q16 + 4375q15 + 3283q14 + 2418q13 + 1748q12 + 1234q11 + 852q10 + 570q9 + 371q8 + 231q7 + 139q6 + 78q5 + 42q4 + 20q3 + 9q2 + 3q + 1 k = q114

– J = {1, 2, 4, 5, 7},... ; type of P : A3A1A1 v = q112 + 3q111 + 8q110 + 17q109 + 34q108 + 61q107 + 105q106 + 170q105 + 266q104 + 400q103 + 586q102 + 834q101 + 1162q100 + 1584q99 + 2121q98 + 2791q97 + 3618q96 + 4623q95 +5831q94 +7266q93 +8952q92 +10914q91 +13174q90 +15756q89 +18676q88 + 21956q87 + 25605q86 + 29639q85 + 34058q84 + 38871q83 + 44066q82 + 49643q81 + 55579q80 + 61862q79 + 68457q78 + 75341q77 + 82467q76 + 89800q75 + 97285q74 + 104875q73+112507q72+120126q71+127665q70+135061q69+142248q68+149159q67+ 155730q66+161895q65+167598q64+172775q63+177381q62+181360q61+184679q60+ 187293q59+189186q58+190325q57+190710q56+190325q55+189186q54+187293q53+ 184679q52+181360q51+177381q50+172775q49+167598q48+161895q47+155730q46+ 149159q45+142248q44+135061q43+127665q42+120126q41+112507q40+104875q39+ 97285q38 + 89800q37 + 82467q36 + 75341q35 + 68457q34 + 61862q33 + 55579q32 + 49643q31 + 44066q30 + 38871q29 + 34058q28 + 29639q27 + 25605q26 + 21956q25 + 18676q24 +15756q23 +13174q22 +10914q21 +8952q20 +7266q19 +5831q18 +4623q17 + 3618q16 +2791q15 +2121q14 +1584q13 +1162q12 +834q11 +586q10 +400q9 +266q8 + 170q7 + 105q6 + 61q5 + 34q4 + 17q3 + 8q2 + 3q + 1 k = q112

– J = {1, 3, 4, 5, 6},... ; type of P : A5 v = q105 +3q104 +6q103 +11q102 +19q101 +31q100 +49q99 +74q98 +107q97 +151q96 + 209q95 + 283q94 + 376q93 + 491q92 + 630q91 + 798q90 + 999q89 + 1235q88 + 1509q87 + 1825q86 + 2185q85 + 2593q84 + 3052q83 + 3562q82 + 4123q81 + 4738q80 + 5407q79 + 6129q78 +6903q77 +7725q76 +8591q75 +9500q74 +10448q73 +11427q72 +12431q71 + 13453q70 + 14485q69 + 15521q68 + 16554q67 + 17571q66 + 18563q65 + 19523q64 + 20442q63 + 21311q62 + 22123q61 + 22866q60 + 23532q59 + 24118q58 + 24618q57 + 25024q56 + 25333q55 + 25540q54 + 25642q53 + 25642q52 + 25540q51 + 25333q50 + 25024q49 + 24618q48 + 24118q47 + 23532q46 + 22866q45 + 22123q44 + 21311q43 + 20442q42 + 19523q41 + 18563q40 + 17571q39 + 16554q38 + 15521q37 + 14485q36 + 13453q35 +12431q34 +11427q33 +10448q32 +9500q31 +8591q30 +7725q29 +6903q28 + 6129q27 + 5407q26 + 4738q25 + 4123q24 + 3562q23 + 3052q22 + 2593q21 + 2185q20 + 1825q19 +1509q18 +1235q17 +999q16 +798q15 +630q14 +491q13 +376q12 +283q11 + 209q10 + 151q9 + 107q8 + 74q7 + 49q6 + 31q5 + 19q4 + 11q3 + 6q2 + 3q + 1 k = q105

– J = {1, 3, 4, 6, 7},... ; type of P : A3A2 v = q111 + 3q110 + 7q109 + 15q108 + 29q107 + 51q106 + 86q105 + 138q104 + 212q103 + 316q102 + 458q101 + 646q100 + 892q99 + 1208q98 + 1605q97 + 2099q96 + 2705q95 + 3437q94 + 4312q93 + 5348q92 + 6558q91 + 7960q90 + 9570q89 + 11400q88 + 13462q87 + 15770q86 + 18329q85 + 21145q84 + 24223q83 + 27561q82 + 31153q81 + 34995q80 + 39074q79 + 43372q78 + 47873q77 + 52553q76 + 57382q75 + 62332q74 + 67371q73 + 72457q72 + 77554q71 + 82622q70 + 87615q69 + 92489q68 + 97205q67 + 101713q66 + 105971q65+109941q64+113581q63+116851q62+119724q61+122166q60+124149q59+ 125657q58+126673q57+127181q56+127181q55+126673q54+125657q53+124149q52+ 122166q51+119724q50+116851q49+113581q48+109941q47+105971q46+101713q45+ 97205q44 + 92489q43 + 87615q42 + 82622q41 + 77554q40 + 72457q39 + 67371q38 + 62332q37 + 57382q36 + 52553q35 + 47873q34 + 43372q33 + 39074q32 + 34995q31 + 31153q30 + 27561q29 + 24223q28 + 21145q27 + 18329q26 + 15770q25 + 13462q24 + 11400q23 + 9570q22 + 7960q21 + 6558q20 + 5348q19 + 4312q18 + 3437q17 + 2705q16 + 2099q15 + 1605q14 + 1208q13 + 892q12 + 646q11 + 458q10 + 316q9 + 212q8 + 138q7 + 138APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

86q6 + 51q5 + 29q4 + 15q3 + 7q2 + 3q + 1 k = q111

– J = {2, 3, 4, 5, 7},... ; type of P : D4A1 v = q107 + 3q106 + 7q105 + 13q104 + 24q103 + 40q102 + 65q101 + 99q100 + 148q99 + 212q98 +299q97 +409q96 +552q95 +728q94 +948q93 +1212q92 +1533q91 +1911q90 + 2358q89 + 2874q88 + 3471q87 + 4149q86 + 4918q85 + 5778q84 + 6735q83 + 7789q82 + 8943q81 + 10197q80 + 11548q79 + 12996q78 + 14532q77 + 16156q76 + 17854q75 + 19626q74 + 21451q73 + 23329q72 + 25234q71 + 27166q70 + 29096q69 + 31024q68 + 32916q67 + 34772q66 + 36559q65 + 38277q64 + 39893q63 + 41407q62 + 42788q61 + 44036q60 + 45126q59 + 46058q58 + 46812q57 + 47388q56 + 47773q55 + 47967q54 + 47967q53 + 47773q52 + 47388q51 + 46812q50 + 46058q49 + 45126q48 + 44036q47 + 42788q46 + 41407q45 + 39893q44 + 38277q43 + 36559q42 + 34772q41 + 32916q40 + 31024q39 + 29096q38 + 27166q37 + 25234q36 + 23329q35 + 21451q34 + 19626q33 + 17854q32 + 16156q31 + 14532q30 + 12996q29 + 11548q28 + 10197q27 + 8943q26 + 7789q25 + 6735q24 + 5778q23 + 4918q22 + 4149q21 + 3471q20 + 2874q19 + 2358q18 + 1911q17 + 1533q16 + 1212q15 + 948q14 + 728q13 + 552q12 + 409q11 + 299q10 + 212q9 + 148q8 + 99q7 + 65q6 + 40q5 + 24q4 + 13q3 + 7q2 + 3q + 1 k = q107

– J = {1, 2, 3, 4, 5, 6}; type of P : E6 v = q84 + 2q83 + 3q82 + 4q81 + 5q80 + 7q79 + 10q78 + 13q77 + 16q76 + 20q75 + 25q74 + 31q73 + 38q72 + 45q71 + 52q70 + 61q69 + 71q68 + 82q67 + 94q66 + 106q65 + 118q64 + 132q63 + 147q62 + 162q61 + 177q60 + 192q59 + 207q58 + 224q57 + 241q56 + 256q55 + 270q54 + 284q53 + 298q52 + 312q51 + 325q50 + 335q49 + 344q48 + 353q47 + 361q46 + 367q45 + 371q44 + 372q43 + 372q42 + 372q41 + 371q40 + 367q39 + 361q38 + 353q37 + 344q36 + 335q35 + 325q34 + 312q33 + 298q32 + 284q31 + 270q30 + 256q29 + 241q28 + 224q27 + 207q26 + 192q25 + 177q24 + 162q23 + 147q22 + 132q21 + 118q20 + 106q19 + 94q18 + 82q17 + 71q16 + 61q15 + 52q14 + 45q13 + 38q12 + 31q11 + 25q10 + 20q9 + 16q8 + 13q7 + 10q6 + 7q5 + 5q4 + 4q3 + 3q2 + 2q + 1 k = q84

– J = {1, 2, 3, 4, 5, 7},... ; type of P : D5A1 v = q99 + 2q98 + 4q97 + 6q96 + 10q95 + 15q94 + 23q93 + 32q92 + 45q91 + 60q90 + 81q89 + 105q88 + 136q87 + 171q86 + 214q85 + 263q84 + 322q83 + 388q82 + 464q81 + 548q80 +643q79 +748q78 +864q77 +990q76 +1125q75 +1271q74 +1426q73 +1592q72 + 1764q71 + 1945q70 + 2129q69 + 2322q68 + 2516q67 + 2716q66 + 2912q65 + 3111q64 + 3303q63 + 3496q62 + 3679q61 + 3858q60 + 4022q59 + 4179q58 + 4320q57 + 4451q56 + 4563q55 + 4660q54 + 4736q53 + 4796q52 + 4836q51 + 4857q50 + 4857q49 + 4836q48 + 4796q47 + 4736q46 + 4660q45 + 4563q44 + 4451q43 + 4320q42 + 4179q41 + 4022q40 + 3858q39 + 3679q38 + 3496q37 + 3303q36 + 3111q35 + 2912q34 + 2716q33 + 2516q32 + 2322q31 + 2129q30 + 1945q29 + 1764q28 + 1592q27 + 1426q26 + 1271q25 + 1125q24 + 990q23 + 864q22 + 748q21 + 643q20 + 548q19 + 464q18 + 388q17 + 322q16 + 263q15 + 214q14 + 171q13 + 136q12 + 105q11 + 81q10 + 60q9 + 45q8 + 32q7 + 23q6 + 15q5 + 10q4 + 6q3 + 4q2 + 2q + 1 k = q99

– J = {1, 2, 3, 4, 6, 7},... ; type of P : A4A2 v = q107 +2q106 +4q105 +8q104 +14q103 +23q102 +37q101 +56q100 +82q99 +118q98 + 165q97 + 225q96 + 302q95 + 398q94 + 515q93 + 659q92 + 831q91 + 1034q90 + 1273q89 + 1551q88 + 1869q87 + 2233q86 + 2644q85 + 3103q84 + 3613q83 + 4177q82 + 4792q81 + 5460q80 + 6181q79 + 6951q78 + 7769q77 + 8634q76 + 9539q75 + 10479q74 + 11452q73 + 12449q72 + 13463q71 + 14489q70 + 15518q69 + 16538q68 + 17546q67 + 18531q66 + 19482q65 + 20392q64 + 21254q63 + 22054q62 + 22789q61 + 23452q60 + 24032q59 + 24524q58 + 24927q57 + 25231q56 + 25435q55 + 25540q54 + 25540q53 + 25435q52 + 25231q51 + 24927q50 + 24524q49 + 24032q48 + 23452q47 + 22789q46 + 22054q45 + 21254q44 + 20392q43 + 19482q42 + 18531q41 + 17546q40 + 16538q39 + 15518q38 + 14489q37+13463q36+12449q35+11452q34+10479q33+9539q32+8634q31+7769q30+ 6951q29 + 6181q28 + 5460q27 + 4792q26 + 4177q25 + 3613q24 + 3103q23 + 2644q22 + A.1. THE CASE J W0 = J 139

2233q21 +1869q20 +1551q19 +1273q18 +1034q17 +831q16 +659q15 +515q14 +398q13 + 302q12 +225q11 +165q10 +118q9 +82q8 +56q7 +37q6 +23q5 +14q4 +8q3 +4q2 +2q+1 k = q107

– J = {1, 2, 3, 4, 6, 8},... ; type of P : A4A1A1 v = q108 + 2q107 + 5q106 + 9q105 + 17q104 + 28q103 + 46q102 + 70q101 + 105q100 + 151q99 + 214q98 + 294q97 + 398q96 + 527q95 + 688q94 + 884q93 + 1121q92 + 1403q91 + 1735q90 + 2123q89 + 2570q88 + 3083q87 + 3663q86 + 4317q85 + 5043q84 + 5850q83 + 6732q82 +7697q81 +8736q80 +9856q79 +11045q78 +12309q77 +13633q76 +15019q75 + 16451q74 + 17929q73 + 19435q72 + 20966q71 + 22504q70 + 24041q69 + 25561q68 + 27054q67 + 28505q66 + 29900q65 + 31228q64 + 32472q63 + 33625q62 + 34670q61 + 35603q60 + 36405q59 + 37078q58 + 37604q57 + 37989q56 + 38217q55 + 38298q54 + 38217q53 + 37989q52 + 37604q51 + 37078q50 + 36405q49 + 35603q48 + 34670q47 + 33625q46 + 32472q45 + 31228q44 + 29900q43 + 28505q42 + 27054q41 + 25561q40 + 24041q39 + 22504q38 + 20966q37 + 19435q36 + 17929q35 + 16451q34 + 15019q33 + 13633q32 +12309q31 +11045q30 +9856q29 +8736q28 +7697q27 +6732q26 +5850q25 + 5043q24 + 4317q23 + 3663q22 + 3083q21 + 2570q20 + 2123q19 + 1735q18 + 1403q17 + 1121q16 + 884q15 + 688q14 + 527q13 + 398q12 + 294q11 + 214q10 + 151q9 + 105q8 + 70q7 + 46q6 + 28q5 + 17q4 + 9q3 + 5q2 + 2q + 1 k = q108

– J = {1, 2, 3, 5, 6, 7},... ; type of P : A3A2A1 v = q110 + 2q109 + 5q108 + 10q107 + 19q106 + 32q105 + 54q104 + 84q103 + 128q102 + 188q101 +270q100 +376q99 +516q98 +692q97 +913q96 +1186q95 +1519q94 +1918q93 + 2394q92 + 2954q91 + 3604q90 + 4356q89 + 5214q88 + 6186q87 + 7276q86 + 8494q85 + 9835q84 + 11310q83 + 12913q82 + 14648q81 + 16505q80 + 18490q79 + 20584q78 + 22788q77 + 25085q76 + 27468q75 + 29914q74 + 32418q73 + 34953q72 + 37504q71 + 40050q70 + 42572q69 + 45043q68 + 47446q67 + 49759q66 + 51954q65 + 54017q64 + 55924q63 + 57657q62 + 59194q61 + 60530q60 + 61636q59 + 62513q58 + 63144q57 + 63529q56 + 63652q55 + 63529q54 + 63144q53 + 62513q52 + 61636q51 + 60530q50 + 59194q49 + 57657q48 + 55924q47 + 54017q46 + 51954q45 + 49759q44 + 47446q43 + 45043q42 + 42572q41 + 40050q40 + 37504q39 + 34953q38 + 32418q37 + 29914q36 + 27468q35 + 25085q34 + 22788q33 + 20584q32 + 18490q31 + 16505q30 + 14648q29 + 12913q28 + 11310q27 + 9835q26 + 8494q25 + 7276q24 + 6186q23 + 5214q22 + 4356q21 + 3604q20 + 2954q19 + 2394q18 + 1918q17 + 1519q16 + 1186q15 + 913q14 + 692q13 + 516q12+376q11+270q10+188q9+128q8+84q7+54q6+32q5+19q4+10q3+5q2+2q+1 k = q110

– J = {1, 2, 3, 5, 6, 8},... ; type of P : A2A2A1A1 v = q112 + 2q111 + 6q110 + 12q109 + 24q108 + 42q107 + 73q106 + 116q105 + 182q104 + 272q103 + 398q102 + 564q101 + 786q100 + 1068q99 + 1429q98 + 1878q97 + 2432q96 + 3104q95 + 3913q94 + 4872q93 + 5998q92 + 7310q91 + 8818q90 + 10542q89 + 12490q88 + 14680q87 + 17111q86 + 19804q85 + 22748q84 + 25958q83 + 29418q82 + 33138q81 + 37089q80 + 41278q79 + 45669q78 + 50256q77 + 54999q76 + 59886q75 + 64867q74 + 69922q73 + 75003q72 + 80076q71 + 85093q70 + 90018q69 + 94802q68 + 99400q67 + 103776q66+107878q65+111674q64+115118q63+118187q62+120830q61+123043q60+ 124780q59+126042q58+126796q57+127058q56+126796q55+126042q54+124780q53+ 123043q52+120830q51+118187q50+115118q49+111674q48+107878q47+103776q46+ 99400q45 + 94802q44 + 90018q43 + 85093q42 + 80076q41 + 75003q40 + 69922q39 + 64867q38 + 59886q37 + 54999q36 + 50256q35 + 45669q34 + 41278q33 + 37089q32 + 33138q31 + 29418q30 + 25958q29 + 22748q28 + 19804q27 + 17111q26 + 14680q25 + 12490q24 + 10542q23 + 8818q22 + 7310q21 + 5998q20 + 4872q19 + 3913q18 + 3104q17 + 2432q16 +1878q15 +1429q14 +1068q13 +786q12 +564q11 +398q10 +272q9 +182q8 + 116q7 + 73q6 + 42q5 + 24q4 + 12q3 + 6q2 + 2q + 1 k = q112

– J = {1, 2, 4, 5, 6, 7},... ; type of P : A5A1 v = q104 + 2q103 + 4q102 + 7q101 + 12q100 + 19q99 + 30q98 + 44q97 + 63q96 + 88q95 + 121q94 + 162q93 + 214q92 + 277q91 + 353q90 + 445q89 + 554q88 + 681q87 + 828q86 + 140APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

997q85 + 1188q84 + 1405q83 + 1647q82 + 1915q81 + 2208q80 + 2530q79 + 2877q78 + 3252q77 + 3651q76 + 4074q75 + 4517q74 + 4983q73 + 5465q72 + 5962q71 + 6469q70 + 6984q69 + 7501q68 + 8020q67 + 8534q66 + 9037q65 + 9526q64 + 9997q63 + 10445q62 + 10866q61 + 11257q60 + 11609q59 + 11923q58 + 12195q57 + 12423q56 + 12601q55 + 12732q54 + 12808q53 + 12834q52 + 12808q51 + 12732q50 + 12601q49 + 12423q48 + 12195q47 + 11923q46 + 11609q45 + 11257q44 + 10866q43 + 10445q42 + 9997q41 + 9526q40 + 9037q39 + 8534q38 + 8020q37 + 7501q36 + 6984q35 + 6469q34 + 5962q33 + 5465q32 + 4983q31 + 4517q30 + 4074q29 + 3651q28 + 3252q27 + 2877q26 + 2530q25 + 2208q24 + 1915q23 + 1647q22 + 1405q21 + 1188q20 + 997q19 + 828q18 + 681q17 + 554q16 +445q15 +353q14 +277q13 +214q12 +162q11 +121q10 +88q9 +63q8 +44q7 + 30q6 + 19q5 + 12q4 + 7q3 + 4q2 + 2q + 1 k = q104

– J = {1, 3, 4, 5, 6, 7},... ; type of P : A6 v = q99 +2q98 +3q97 +5q96 +8q95 +12q94 +18q93 +26q92 +35q91 +47q90 +63q89 + 82q88 + 105q87 + 133q86 + 165q85 + 203q84 + 248q83 + 299q82 + 356q81 + 421q80 + 493q79 +573q78 +662q77 +758q76 +860q75 +971q74 +1090q73 +1215q72 +1347q71 + 1484q70 + 1624q69 + 1769q68 + 1919q67 + 2069q66 + 2219q65 + 2369q64 + 2516q63 + 2660q62 + 2802q61 + 2936q60 + 3061q59 + 3179q58 + 3288q57 + 3385q56 + 3472q55 + 3545q54 + 3602q53 + 3647q52 + 3679q51 + 3694q50 + 3694q49 + 3679q48 + 3647q47 + 3602q46 + 3545q45 + 3472q44 + 3385q43 + 3288q42 + 3179q41 + 3061q40 + 2936q39 + 2802q38 + 2660q37 + 2516q36 + 2369q35 + 2219q34 + 2069q33 + 1919q32 + 1769q31 + 1624q30 +1484q29 +1347q28 +1215q27 +1090q26 +971q25 +860q24 +758q23 +662q22 + 573q21 + 493q20 + 421q19 + 356q18 + 299q17 + 248q16 + 203q15 + 165q14 + 133q13 + 105q12 + 82q11 + 63q10 + 47q9 + 35q8 + 26q7 + 18q6 + 12q5 + 8q4 + 5q3 + 3q2 + 2q + 1 k = q99

– J = {1, 3, 4, 6, 7, 8},... ; type of P : A3A3 v = q108 + 2q107 + 4q106 + 8q105 + 15q104 + 24q103 + 39q102 + 60q101 + 89q100 + 128q99 + 181q98 + 248q97 + 335q96 + 444q95 + 578q94 + 742q93 + 941q92 + 1176q91 + 1453q90 + 1778q89 + 2151q88 + 2578q87 + 3063q86 + 3608q85 + 4213q84 + 4886q83 + 5622q82 +6424q81 +7291q80 +8224q79 +9214q78 +10266q77 +11370q76 +12522q75 + 13715q74 + 14946q73 + 16199q72 + 17472q71 + 18754q70 + 20032q69 + 21296q68 + 22540q67 + 23747q66 + 24906q65 + 26012q64 + 27048q63 + 28005q62 + 28876q61 + 29652q60 + 30318q59 + 30878q58 + 31318q57 + 31635q56 + 31826q55 + 31894q54 + 31826q53 + 31635q52 + 31318q51 + 30878q50 + 30318q49 + 29652q48 + 28876q47 + 28005q46 + 27048q45 + 26012q44 + 24906q43 + 23747q42 + 22540q41 + 21296q40 + 20032q39 + 18754q38 + 17472q37 + 16199q36 + 14946q35 + 13715q34 + 12522q33 + 11370q32 + 10266q31 + 9214q30 + 8224q29 + 7291q28 + 6424q27 + 5622q26 + 4886q25 + 4213q24 + 3608q23 + 3063q22 + 2578q21 + 2151q20 + 1778q19 + 1453q18 + 1176q17 + 941q16 + 742q15 + 578q14 + 444q13 + 335q12 + 248q11 + 181q10 + 128q9 + 89q8 + 60q7 + 39q6 + 24q5 + 15q4 + 8q3 + 4q2 + 2q + 1 k = q108

– J = {2, 3, 4, 5, 6, 7}; type of P : D6 v = q90 + 2q89 + 3q88 + 4q87 + 6q86 + 8q85 + 12q84 + 16q83 + 21q82 + 26q81 + 34q80 + 42q79 +53q78 +64q77 +77q76 +90q75 +108q74 +126q73 +147q72 +168q71 +192q70 + 216q69 + 245q68 + 274q67 + 304q66 + 334q65 + 367q64 + 400q63 + 435q62 + 470q61 + 503q60 + 536q59 + 570q58 + 604q57 + 635q56 + 666q55 + 692q54 + 718q53 + 742q52 + 766q51 + 782q50 + 798q49 + 808q48 + 818q47 + 823q46 + 828q45 + 823q44 + 818q43 + 808q42 + 798q41 + 782q40 + 766q39 + 742q38 + 718q37 + 692q36 + 666q35 + 635q34 + 604q33 + 570q32 + 536q31 + 503q30 + 470q29 + 435q28 + 400q27 + 367q26 + 334q25 + 304q24 + 274q23 + 245q22 + 216q21 + 192q20 + 168q19 + 147q18 + 126q17 + 108q16 + 90q15 + 77q14 + 64q13 + 53q12 + 42q11 + 34q10 + 26q9 + 21q8 + 16q7 + 12q6 + 8q5 + 6q4 + 4q3 + 3q2 + 2q + 1 k = q90

– J = {2, 3, 4, 5, 7, 8}; type of P : D4A2 v = q105 + 2q104 + 4q103 + 7q102 + 13q101 + 20q100 + 32q99 + 47q98 + 69q97 + 96q96 + A.1. THE CASE J W0 = J 141

134q95 + 179q94 + 239q93 + 310q92 + 399q91 + 503q90 + 631q89 + 777q88 + 950q87 + 1147q86 + 1374q85 + 1628q84 + 1916q83 + 2234q82 + 2585q81 + 2970q80 + 3388q79 + 3839q78 + 4321q77 + 4836q76 + 5375q75 + 5945q74 + 6534q73 + 7147q72 + 7770q71 + 8412q70+9052q69+9702q68+10342q67+10980q66+11594q65+12198q64+12767q63+ 13312q62 + 13814q61 + 14281q60 + 14693q59 + 15062q58 + 15371q57 + 15625q56 + 15816q55 + 15947q54 + 16010q53 + 16010q52 + 15947q51 + 15816q50 + 15625q49 + 15371q48 + 15062q47 + 14693q46 + 14281q45 + 13814q44 + 13312q43 + 12767q42 + 12198q41 +11594q40 +10980q39 +10342q38 +9702q37 +9052q36 +8412q35 +7770q34 + 7147q33 + 6534q32 + 5945q31 + 5375q30 + 4836q29 + 4321q28 + 3839q27 + 3388q26 + 2970q25 + 2585q24 + 2234q23 + 1916q22 + 1628q21 + 1374q20 + 1147q19 + 950q18 + 777q17 + 631q16 + 503q15 + 399q14 + 310q13 + 239q12 + 179q11 + 134q10 + 96q9 + 69q8 + 47q7 + 32q6 + 20q5 + 13q4 + 7q3 + 4q2 + 2q + 1 k = q105

– J = {1, 2, 3, 4, 5, 6, 7}; type of P : E7 v = q57 + q56 + q55 + q54 + q53 + q52 + 2q51 + 2q50 + 2q49 + 2q48 + 3q47 + 3q46 + 4q45 + 4q44 + 4q43 + 4q42 + 5q41 + 5q40 + 6q39 + 6q38 + 6q37 + 6q36 + 7q35 + 7q34 + 7q33 + 7q32 + 7q31 + 7q30 + 8q29 + 8q28 + 7q27 + 7q26 + 7q25 + 7q24 + 7q23 + 7q22 + 6q21 + 6q20 + 6q19 + 6q18 + 5q17 + 5q16 + 4q15 + 4q14 + 4q13 + 4q12 + 3q11 + 3q10 + 2q9 + 2q8 + 2q7 + 2q6 + q5 + q4 + q3 + q2 + q + 1 k = q57

– J = {1, 2, 3, 4, 5, 6, 8}; type of P : D6A1 v = q83 +q82 +2q81 +2q80 +3q79 +4q78 +6q77 +7q76 +9q75 +11q74 +14q73 +17q72 + 21q71 +24q70 +28q69 +33q68 +38q67 +44q66 +50q65 +56q64 +62q63 +70q62 +77q61 + 85q60 + 92q59 + 100q58 + 107q57 + 117q56 + 124q55 + 132q54 + 138q53 + 146q52 + 152q51 + 160q50 + 165q49 + 170q48 + 174q47 + 179q46 + 182q45 + 185q44 + 186q43 + 186q42 + 186q41 + 186q40 + 185q39 + 182q38 + 179q37 + 174q36 + 170q35 + 165q34 + 160q33 + 152q32 + 146q31 + 138q30 + 132q29 + 124q28 + 117q27 + 107q26 + 100q25 + 92q24 +85q23 +77q22 +70q21 +62q20 +56q19 +50q18 +44q17 +38q16 +33q15 +28q14 + 24q13 + 21q12 + 17q11 + 14q10 + 11q9 + 9q8 + 7q7 + 6q6 + 4q5 + 3q4 + 2q3 + 2q2 + q + 1 k = q83

– J = {1, 2, 3, 4, 5, 7, 8}; type of P : D5A2 v = q97 + q96 + 2q95 + 3q94 + 5q93 + 7q92 + 11q91 + 14q90 + 20q89 + 26q88 + 35q87 + 44q86 + 57q85 + 70q84 + 87q83 + 106q82 + 129q81 + 153q80 + 182q79 + 213q78 + 248q77 + 287q76 + 329q75 + 374q74 + 422q73 + 475q72 + 529q71 + 588q70 + 647q69 + 710q68 +772q67 +840q66 +904q65 +972q64 +1036q63 +1103q62 +1164q61 +1229q60 + 1286q59 + 1343q58 + 1393q57 + 1443q56 + 1484q55 + 1524q54 + 1555q53 + 1581q52 + 1600q51 + 1615q50 + 1621q49 + 1621q48 + 1615q47 + 1600q46 + 1581q45 + 1555q44 + 1524q43 + 1484q42 + 1443q41 + 1393q40 + 1343q39 + 1286q38 + 1229q37 + 1164q36 + 1103q35 + 1036q34 + 972q33 + 904q32 + 840q31 + 772q30 + 710q29 + 647q28 + 588q27 + 529q26 + 475q25 + 422q24 + 374q23 + 329q22 + 287q21 + 248q20 + 213q19 + 182q18 + 153q17 + 129q16 + 106q15 + 87q14 + 70q13 + 57q12 + 44q11 + 35q10 + 26q9 + 20q8 + 14q7 + 11q6 + 7q5 + 5q4 + 3q3 + 2q2 + q + 1 k = q97

– J = {1, 2, 3, 4, 6, 7, 8}; type of P : A4A3 v = q104 + q103 + 2q102 + 4q101 + 7q100 + 10q99 + 16q98 + 23q97 + 33q96 + 46q95 + 63q94 + 83q93 + 110q92 + 142q91 + 180q90 + 227q89 + 282q88 + 345q87 + 419q86 + 505q85 +600q84 +709q83 +830q82 +964q81 +1110q80 +1273q79 +1445q78 +1632q77 + 1831q76 + 2043q75 + 2263q74 + 2497q73 + 2736q72 + 2983q71 + 3236q70 + 3494q69 + 3750q68 + 4009q67 + 4265q66 + 4514q65 + 4758q64 + 4994q63 + 5216q62 + 5424q61 + 5620q60 + 5794q59 + 5951q58 + 6087q57 + 6200q56 + 6286q55 + 6354q54 + 6391q53 + 6404q52 + 6391q51 + 6354q50 + 6286q49 + 6200q48 + 6087q47 + 5951q46 + 5794q45 + 5620q44 + 5424q43 + 5216q42 + 4994q41 + 4758q40 + 4514q39 + 4265q38 + 4009q37 + 3750q36 + 3494q35 + 3236q34 + 2983q33 + 2736q32 + 2497q31 + 2263q30 + 2043q29 + 1831q28 + 1632q27 + 1445q26 + 1273q25 + 1110q24 + 964q23 + 830q22 + 709q21 + 600q20 + 505q19 + 419q18 + 345q17 + 282q16 + 227q15 + 180q14 + 142q13 + 110q12 + 142APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

83q11 + 63q10 + 46q9 + 33q8 + 23q7 + 16q6 + 10q5 + 7q4 + 4q3 + 2q2 + q + 1 k = q104

– J = {1, 2, 3, 5, 6, 7, 8}; type of P : A4A2A1 v = q106 + q105 + 3q104 + 5q103 + 9q102 + 14q101 + 23q100 + 33q99 + 49q98 + 69q97 + 96q96 + 129q95 + 173q94 + 225q93 + 290q92 + 369q91 + 462q90 + 572q89 + 701q88 + 850q87 + 1019q86 + 1214q85 + 1430q84 + 1673q83 + 1940q82 + 2237q81 + 2555q80 + 2905q79 + 3276q78 + 3675q77 + 4094q76 + 4540q75 + 4999q74 + 5480q73 + 5972q72 + 6477q71 + 6986q70 + 7503q69 + 8015q68 + 8523q67 + 9023q66 + 9508q65 + 9974q64 + 10418q63 + 10836q62 + 11218q61 + 11571q60 + 11881q59 + 12151q58 + 12373q57 + 12554q56 + 12677q55 + 12758q54 + 12782q53 + 12758q52 + 12677q51 + 12554q50 + 12373q49 + 12151q48 + 11881q47 + 11571q46 + 11218q45 + 10836q44 + 10418q43 + 9974q42 + 9508q41 + 9023q40 + 8523q39 + 8015q38 + 7503q37 + 6986q36 + 6477q35 + 5972q34 + 5480q33 + 4999q32 + 4540q31 + 4094q30 + 3675q29 + 3276q28 + 2905q27 + 2555q26 + 2237q25 + 1940q24 + 1673q23 + 1430q22 + 1214q21 + 1019q20 + 850q19 + 701q18 + 572q17 + 462q16 + 369q15 + 290q14 + 225q13 + 173q12 + 129q11 + 96q10 + 69q9 + 49q8 + 33q7 + 23q6 + 14q5 + 9q4 + 5q3 + 3q2 + q + 1 k = q106

– J = {1, 2, 4, 5, 6, 7, 8}; type of P : A6A1 v = q98 + q97 + 2q96 + 3q95 + 5q94 + 7q93 + 11q92 + 15q91 + 20q90 + 27q89 + 36q88 + 46q87 +59q86 +74q85 +91q84 +112q83 +136q82 +163q81 +193q80 +228q79 +265q78 + 308q77 + 354q76 + 404q75 + 456q74 + 515q73 + 575q72 + 640q71 + 707q70 + 777q69 + 847q68 + 922q67 + 997q66 + 1072q65 + 1147q64 + 1222q63 + 1294q62 + 1366q61 + 1436q60 + 1500q59 + 1561q58 + 1618q57 + 1670q56 + 1715q55 + 1757q54 + 1788q53 + 1814q52 + 1833q51 + 1846q50 + 1848q49 + 1846q48 + 1833q47 + 1814q46 + 1788q45 + 1757q44 + 1715q43 + 1670q42 + 1618q41 + 1561q40 + 1500q39 + 1436q38 + 1366q37 + 1294q36 +1222q35 +1147q34 +1072q33 +997q32 +922q31 +847q30 +777q29 +707q28 + 640q27 + 575q26 + 515q25 + 456q24 + 404q23 + 354q22 + 308q21 + 265q20 + 228q19 + 193q18 + 163q17 + 136q16 + 112q15 + 91q14 + 74q13 + 59q12 + 46q11 + 36q10 + 27q9 + 20q8 + 15q7 + 11q6 + 7q5 + 5q4 + 3q3 + 2q2 + q + 1 k = q98

– J = {1, 3, 4, 5, 6, 7, 8}; type of P : A7 v = q92 +q91 +q90 +2q89 +3q88 +4q87 +6q86 +8q85 +10q84 +13q83 +17q82 +21q81 + 26q80+32q79+38q78+46q77+55q76+64q75+74q74+86q73+98q72+112q71+127q70+ 142q69 + 157q68 + 175q67 + 193q66 + 211q65 + 230q64 + 249q63 + 267q62 + 287q61 + 307q60 + 325q59 + 343q58 + 361q57 + 377q56 + 393q55 + 409q54 + 421q53 + 432q52 + 443q51 + 452q50 + 458q49 + 464q48 + 466q47 + 466q46 + 466q45 + 464q44 + 458q43 + 452q42 + 443q41 + 432q40 + 421q39 + 409q38 + 393q37 + 377q36 + 361q35 + 343q34 + 325q33 + 307q32 + 287q31 + 267q30 + 249q29 + 230q28 + 211q27 + 193q26 + 175q25 + 157q24 + 142q23 + 127q22 + 112q21 + 98q20 + 86q19 + 74q18 + 64q17 + 55q16 + 46q15 + 38q14+32q13+26q12+21q11+17q10+13q9+10q8+8q7+6q6+4q5+3q4+2q3+q2+q+1 k = q92

– J = {2, 3, 4, 5, 6, 7, 8}; type of P : D7 v = q78 + q77 + q76 + q75 + 2q74 + 2q73 + 3q72 + 4q71 + 5q70 + 5q69 + 7q68 + 8q67 + 10q66 + 11q65 + 13q64 + 14q63 + 17q62 + 19q61 + 21q60 + 23q59 + 26q58 + 28q57 + 31q56 + 34q55 + 36q54 + 38q53 + 41q52 + 44q51 + 46q50 + 49q49 + 50q48 + 52q47 + 54q46 + 57q45 + 57q44 + 59q43 + 59q42 + 60q41 + 60q40 + 62q39 + 60q38 + 60q37 + 59q36 +59q35 +57q34 +57q33 +54q32 +52q31 +50q30 +49q29 +46q28 +44q27 +41q26 + 38q25 +36q24 +34q23 +31q22 +28q21 +26q20 +23q19 +21q18 +19q17 +17q16 +14q15 + 13q14 +11q13 +10q12 +8q11 +7q10 +5q9 +5q8 +4q7 +3q6 +2q5 +2q4 +q3 +q2 +q +1 k = q78

F8 – J = {1}, {2}, {3}, {4}; type of P : A1 v = q23 +3q22 +6q21 +10q20 +15q19 +21q18 +27q17 +33q16 +38q15 +42q14 +45q13 + 47q12 + 47q11 + 45q10 + 42q9 + 38q8 + 33q7 + 27q6 + 21q5 + 15q4 + 10q3 + 6q2 + 3q + 1 k = q23 A.2. THE CASE J W0 6= J 143

– J = {1, 2}, {3, 4}; type of P : A2 v = q21 + 2q20 + 3q19 + 5q18 + 7q17 + 9q16 + 11q15 + 13q14 + 14q13 + 15q12 + 16q11 + 16q10 + 15q9 + 14q8 + 13q7 + 11q6 + 9q5 + 7q4 + 5q3 + 3q2 + 2q + 1 k = q21

– J = {1, 3}, {1, 4}, {2, 4}; type of P : A1A1 v = q22 +2q21 +4q20 +6q19 +9q18 +12q17 +15q16 +18q15 +20q14 +22q13 +23q12 + 24q11 + 23q10 + 22q9 + 20q8 + 18q7 + 15q6 + 12q5 + 9q4 + 6q3 + 4q2 + 2q + 1 k = q22

– J = {2, 3}; type of P : B2 v = q20 + 2q19 + 3q18 + 4q17 + 6q16 + 8q15 + 9q14 + 10q13 + 11q12 + 12q11 + 12q10 + 12q9 + 11q8 + 10q7 + 9q6 + 8q5 + 6q4 + 4q3 + 3q2 + 2q + 1 k = q20

– J = {1, 2, 3}, {2, 3, 4}; type of P : B3 v = q15 +q14 +q13 +q12 +2q11 +2q10 +2q9 +2q8 +2q7 +2q6 +2q5 +2q4 +q3 +q2 +q+1 k = q15

– J = {1, 2, 4}, {1, 3, 4}; type of P : A2A1 v = q20 + q19 + 2q18 + 3q17 + 4q16 + 5q15 + 6q14 + 7q13 + 7q12 + 8q11 + 8q10 + 8q9 + 7q8 + 7q7 + 6q6 + 5q5 + 4q4 + 3q3 + 2q2 + q + 1 k = q20

G2 – J = {1}, {2}; type of P : A1 v = q5 + q4 + q3 + q2 + q + 1 k = q5

A.2 The case J w0 6= J

A2 – J = {1}, {2}; type of P : A1 v = q2 + q + 1 k = q(q + 1)

A3 – J = {1}, {3}; type of P : A1 v = q5 + 2q4 + 3q3 + 3q2 + 2q + 1 k = q4(q + 1)

– J = {1, 2}, {2, 3}; type of P : A2 v = q3 + q2 + q + 1 k = q(q2 + q + 1)

A4 – J = {1}, {2}, {3}, {4}; type of P : A1 v = q9 + 3q8 + 6q7 + 9q6 + 11q5 + 11q4 + 9q3 + 6q2 + 3q + 1 k = q8(q + 1)

– J = {1, 2}, {3, 4}; type of P : A2 v = q7 + 2q6 + 3q5 + 4q4 + 4q3 + 3q2 + 2q + 1 k = q4(q3 + 2q2 + 2q + 1)

– J = {1, 3}, {2, 4}; type of P : A1A1 v = q8 + 2q7 + 4q6 + 5q5 + 6q4 + 5q3 + 4q2 + 2q + 1 k = q6(q2 + 2q + 1)

– J = {1, 2, 3}, {2, 3, 4}; type of P : A3 v = q4 + q3 + q2 + q + 1 k = q(q3 + q2 + q + 1)

– J = {1, 2, 4}, {1, 3, 4}; type of P : A2A1 v = q6 + q5 + 2q4 + 2q3 + 2q2 + q + 1 k = q4(q2 + q + 1) 144APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

A5 – J = {1}, {2}, {4}, {5}; type of P : A1 v = q14 + 4q13 + 10q12 + 19q11 + 30q10 + 41q9 + 49q8 + 52q7 + 49q6 + 41q5 + 30q4 + 19q3 + 10q2 + 4q + 1 k = q13(q + 1)

– J = {1, 2}, {4, 5}; type of P : A2 v = q12 + 3q11 + 6q10 + 10q9 + 14q8 + 17q7 + 18q6 + 17q5 + 14q4 + 10q3 + 6q2 + 3q + 1 k = q9(q3 + 2q2 + 2q + 1)

– J = {1, 3}, {3, 5}; type of P : A1A1 v = q13 + 3q12 + 7q11 + 12q10 + 18q9 + 23q8 + 26q7 + 26q6 + 23q5 + 18q4 + 12q3 + 7q2 + 3q + 1 k = q12(q + 1)

– J = {1, 4}, {2, 5}; type of P : A1A1 v = q13 + 3q12 + 7q11 + 12q10 + 18q9 + 23q8 + 26q7 + 26q6 + 23q5 + 18q4 + 12q3 + 7q2 + 3q + 1 k = q11(q2 + 2q + 1)

– J = {2, 3}, {3, 4}; type of P : A2 v = q12 + 3q11 + 6q10 + 10q9 + 14q8 + 17q7 + 18q6 + 17q5 + 14q4 + 10q3 + 6q2 + 3q + 1 k = q10(q2 + q + 1)

– J = {1, 2, 3}, {3, 4, 5}; type of P : A3 v = q9 + 2q8 + 3q7 + 4q6 + 5q5 + 5q4 + 4q3 + 3q2 + 2q + 1 k = q4(q5 + 2q4 + 3q3 + 3q2 + 2q + 1)

– J = {1, 2, 4}, {1, 2, 5}, {1, 4, 5}, {2, 4, 5}; type of P : A2A1 v = q11 + 2q10 + 4q9 + 6q8 + 8q7 + 9q6 + 9q5 + 8q4 + 6q3 + 4q2 + 2q + 1 k = q9(q2 + q + 1)

– J = {1, 3, 4}, {2, 3, 5}; type of P : A2A1 v = q11 + 2q10 + 4q9 + 6q8 + 8q7 + 9q6 + 9q5 + 8q4 + 6q3 + 4q2 + 2q + 1 k = q8(q3 + 2q2 + 2q + 1)

– J = {1, 2, 3, 4}, {2, 3, 4, 5}; type of P : A4 v = q5 + q4 + q3 + q2 + q + 1 k = q(q4 + q3 + q3 + q + 1)

– J = {1, 2, 3, 5}, {1, 3, 4, 5}; type of P : A3A1 v = q8 + q7 + 2q6 + 2q5 + 3q4 + 2q3 + 2q2 + q + 1 k = q4(q4 + q3 + 2q2 + q + 1)

D5 – J = {4}, {5}; type of P : A1 v = q19 + 4q18 + 10q17 + 20q16 + 34q15 + 51q14 + 69q13 + 86q12 + 99q11 + 106q10 + 106q9 + 99q8 + 86q7 + 69q6 + 51q5 + 34q4 + 20q3 + 10q2 + 4q + 1 k = q18(q + 1)

– J = {1, 4}, {1, 5}, {2, 4}, {2, 5}; type of P : A1A1 v = q18 + 3q17 + 7q16 + 13q15 + 21q14 + 30q13 + 39q12 + 47q11 + 52q10 + 54q9 + 52q8 + 47q7 + 39q6 + 30q5 + 21q4 + 13q3 + 7q2 + 3q + 1 k = q17(q + 1)

– J = {3, 4}, {3, 5}; type of P : A2 v = q17 + 3q16 + 6q15 + 11q14 + 17q13 + 23q12 + 29q11 + 34q10 + 36q9 + 36q8 + 34q7 + 29q6 + 23q5 + 17q4 + 11q3 + 6q2 + 3q + 1 k = q15(q2 + q + 1)

– J = {1, 2, 4}, {1, 2, 5}; type of P : A2A1 v = q16 + 2q15 + 4q14 + 7q13 + 10q12 + 13q11 + 16q10 + 18q9 + 18q8 + 18q7 + 16q6 + 13q5 + 10q4 + 7q3 + 4q2 + 2q + 1 k = q15(q + 1) A.2. THE CASE J W0 6= J 145

– J = {1, 3, 4}, {1, 3, 5}; type of P : A2A1 v = q16 + 2q15 + 4q14 + 7q13 + 10q12 + 13q11 + 16q10 + 18q9 + 18q8 + 18q7 + 16q6 + 13q5 + 10q4 + 7q3 + 4q2 + 2q + 1 k = q14(q2 + q + 1)

– J = {2, 3, 4}, {2, 3, 5}; type of P : A3 v = q14+2q13+3q12+5q11+7q10+8q9+9q8+10q7+9q6+8q5+7q4+5q3+3q2+2q+1 k = q11(q3 + q2 + q + 1)

– J = {1, 2, 3, 4}, {1, 2, 3, 5}; type of P : A4 v = q10 + q9 + q8 + 2q7 + 2q6 + 2q5 + 2q4 + 2q3 + q2 + q + 1 k = q6(q4 + q3 + q2 + q + 1)

E6 – J = {1}, {3}, {5}, {6}; type of P : A1 v = q35 +5q34 +15q33 +35q32 +70q31 +125q30 +204q29 +310q28 +444q27 +604q26 + 785q25 + 980q24 + 1179q23 + 1370q22 + 1541q21 + 1681q20 + 1780q19 + 1831q18 + 1831q17 + 1780q16 + 1681q15 + 1541q14 + 1370q13 + 1179q12 + 980q11 + 785q10 + 604q9 + 444q8 + 310q7 + 204q6 + 125q5 + 70q4 + 35q3 + 15q2 + 5q + 1 k = q34(q + 1)

– J = {1, 2}, {1, 4}, {2, 3}, {2, 5}, {2, 6}, {4, 6}; type of P : A1A1 v = q34 +4q33 +11q32 +24q31 +46q30 +79q29 +125q28 +185q27 +259q26 +345q25 + 440q24 + 540q23 + 639q22 + 731q21 + 810q20 + 871q19 + 909q18 + 922q17 + 909q16 + 871q15 + 810q14 + 731q13 + 639q12 + 540q11 + 440q10 + 345q9 + 259q8 + 185q7 + 125q6 + 79q5 + 46q4 + 24q3 + 11q2 + 4q + 1 k = q33(q + 1)

– J = {1, 3}, {5, 6}; type of P : A2 v = q33 +4q32 +10q31 +21q30 +39q29 +65q28 +100q27 +145q26 +199q25 +260q24 + 326q23 + 394q22 + 459q21 + 517q20 + 565q19 + 599q18 + 616q17 + 616q16 + 599q15 + 565q14 +517q13 +459q12 +394q11 +326q10 +260q9 +199q8 +145q7 +100q6 +65q5 + 39q4 + 21q3 + 10q2 + 4q + 1 k = q30(q3 + 2q2 + 2q + 1)

– J = {1, 5}, {3, 6}; type of P : A1A1 v = q34 +4q33 +11q32 +24q31 +46q30 +79q29 +125q28 +185q27 +259q26 +345q25 + 440q24 + 540q23 + 639q22 + 731q21 + 810q20 + 871q19 + 909q18 + 922q17 + 909q16 + 871q15 + 810q14 + 731q13 + 639q12 + 540q11 + 440q10 + 345q9 + 259q8 + 185q7 + 125q6 + 79q5 + 46q4 + 24q3 + 11q2 + 4q + 1 k = q32(q2 + 2q + 1)

– J = {3, 4}, {4, 5}; type of P : A2 v = q33 +4q32 +10q31 +21q30 +39q29 +65q28 +100q27 +145q26 +199q25 +260q24 + 326q23 + 394q22 + 459q21 + 517q20 + 565q19 + 599q18 + 616q17 + 616q16 + 599q15 + 565q14 +517q13 +459q12 +394q11 +326q10 +260q9 +199q8 +145q7 +100q6 +65q5 + 39q4 + 21q3 + 10q2 + 4q + 1 k = q31(q2 + q + 1)

– J = {1, 2, 3}, {1, 4, 5}, {2, 5, 6}, {3, 4, 6}; type of P : A2A1 v = q32 + 3q31 + 7q30 + 14q29 + 25q28 + 40q27 + 60q26 + 85q25 + 114q24 + 146q23 + 180q22 + 214q21 + 245q20 + 272q19 + 293q18 + 306q17 + 310q16 + 306q15 + 293q14 + 272q13 + 245q12 + 214q11 + 180q10 + 146q9 + 114q8 + 85q7 + 60q6 + 40q5 + 25q4 + 14q3 + 7q2 + 3q + 1 k = q29(q3 + 2q2 + 2q + 1)

– J = {1, 2, 4}, {2, 4, 6}; type of P : A2A1 v = q32 + 3q31 + 7q30 + 14q29 + 25q28 + 40q27 + 60q26 + 85q25 + 114q24 + 146q23 + 180q22 + 214q21 + 245q20 + 272q19 + 293q18 + 306q17 + 310q16 + 306q15 + 293q14 + 272q13 + 245q12 + 214q11 + 180q10 + 146q9 + 114q8 + 85q7 + 60q6 + 40q5 + 25q4 + 14q3 + 7q2 + 3q + 1 k = q31(q + 1) 146APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS

– J = {1, 2, 5}, {2, 3, 6}; type of P : A1A1A1 v = q33 + 3q32 + 8q31 + 16q30 + 30q29 + 49q28 + 76q27 + 109q26 + 150q25 + 195q24 + 245q23 + 295q22 + 344q21 + 387q20 + 423q19 + 448q18 + 461q17 + 461q16 + 448q15 + 423q14 + 387q13 + 344q12 + 295q11 + 245q10 + 195q9 + 150q8 + 109q7 + 76q6 + 49q5 + 30q4 + 16q3 + 8q2 + 3q + 1 k = q31(q2 + 2q + 1)

– J = {1, 3, 4}, {4, 5, 6}; type of P : A3 v = q30 + 3q29 + 6q28 + 11q27 + 19q26 + 29q25 + 41q24 + 56q23 + 73q22 + 90q21 + 107q20 + 124q19 + 138q18 + 148q17 + 155q16 + 158q15 + 155q14 + 148q13 + 138q12 + 124q11 + 107q10 + 90q9 + 73q8 + 56q7 + 41q6 + 29q5 + 19q4 + 11q3 + 6q2 + 3q + 1 k = q25(q5 + 2q4 + 3q3 + 3q2 + 2q + 1)

– J = {1, 3, 5}, {1, 3, 6}, {1, 5, 6}, {3, 5, 6}; type of P : A2A1 v = q32 + 3q31 + 7q30 + 14q29 + 25q28 + 40q27 + 60q26 + 85q25 + 114q24 + 146q23 + 180q22 + 214q21 + 245q20 + 272q19 + 293q18 + 306q17 + 310q16 + 306q15 + 293q14 + 272q13 + 245q12 + 214q11 + 180q10 + 146q9 + 114q8 + 85q7 + 60q6 + 40q5 + 25q4 + 14q3 + 7q2 + 3q + 1 k = q30(q2 + q + 1)

– J = {2, 3, 4}, {2, 4, 5}; type of P : A3 v = q30 + 3q29 + 6q28 + 11q27 + 19q26 + 29q25 + 41q24 + 56q23 + 73q22 + 90q21 + 107q20 + 124q19 + 138q18 + 148q17 + 155q16 + 158q15 + 155q14 + 148q13 + 138q12 + 124q11 + 107q10 + 90q9 + 73q8 + 56q7 + 41q6 + 29q5 + 19q4 + 11q3 + 6q2 + 3q + 1 k = q27(q3 + q2 + q + 1)

– J = {1, 2, 3, 4}, {2, 4, 5, 6}; type of P : A4 v = q26 +2q25 +3q24 +5q23 +8q22 +11q21 +14q20 +18q19 +22q18 +25q17 +28q16 + 31q15 + 32q14 + 32q13 + 32q12 + 31q11 + 28q10 + 25q9 + 22q8 + 18q7 + 14q6 + 11q5 + 8q4 + 5q3 + 3q2 + 2q + 1 k = q19(q7 + 2q6 + 3q5 + 4q4 + 4q3 + 3q2 + 2q + 1)

– J = {1, 2, 3, 5}, {1, 2, 3, 6}, {1, 2, 5, 6}, {2, 3, 5, 6}; type of P : A2A1A1 v = q31 + 2q30 + 5q29 + 9q28 + 16q27 + 24q26 + 36q25 + 49q24 + 65q23 + 81q22 + 99q21 + 115q20 + 130q19 + 142q18 + 151q17 + 155q16 + 155q15 + 151q14 + 142q13 + 130q12 +115q11 +99q10 +81q9 +65q8 +49q7 +36q6 +24q5 +16q4 +9q3 +5q2 +2q +1 k = q29(q2 + q + 1)

– J = {1, 2, 4, 5}, {2, 3, 4, 6}; type of P : A3A1 v = q29 + 2q28 + 4q27 + 7q26 + 12q25 + 17q24 + 24q23 + 32q22 + 41q21 + 49q20 + 58q19 + 66q18 + 72q17 + 76q16 + 79q15 + 79q14 + 76q13 + 72q12 + 66q11 + 58q10 + 49q9 + 41q8 + 32q7 + 24q6 + 17q5 + 12q4 + 7q3 + 4q2 + 2q + 1 k = q25(q4 + 2q3 + 2q2 + 2q + 1)

– J = {1, 3, 4, 5}, {3, 4, 5, 6}; type of P : A4 v = q26 +2q25 +3q24 +5q23 +8q22 +11q21 +14q20 +18q19 +22q18 +25q17 +28q16 + 31q15 + 32q14 + 32q13 + 32q12 + 31q11 + 28q10 + 25q9 + 22q8 + 18q7 + 14q6 + 11q5 + 8q4 + 5q3 + 3q2 + 2q + 1 k = q22(q4 + q3 + q2 + q + 1)

– J = {1, 3, 4, 6}, {1, 4, 5, 6}; type of P : A3A1 v = q29 + 2q28 + 4q27 + 7q26 + 12q25 + 17q24 + 24q23 + 32q22 + 41q21 + 49q20 + 58q19 + 66q18 + 72q17 + 76q16 + 79q15 + 79q14 + 76q13 + 72q12 + 66q11 + 58q10 + 49q9 + 41q8 + 32q7 + 24q6 + 17q5 + 12q4 + 7q3 + 4q2 + 2q + 1 k = q25(q4 + q3 + 2q2 + q + 1)

– J = {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}; type of P : D5 v = q16 + q15 + q14 + q13 + 2q12 + 2q11 + 2q10 + 2q9 + 3q8 + 2q7 + 2q6 + 2q5 + 2q4 + q3 + q2 + q + 1 k = q8(q8 + q7 + q6 + q5 + 2q4 + q3 + q2 + q + 1)

– J = {1, 2, 3, 4, 6}, {1, 2, 4, 5, 6}; type of P : A4A1 v = q25 +q24 +2q23 +3q22 +5q21 +6q20 +8q19 +10q18 +12q17 +13q16 +15q15 +16q14 + A.2. THE CASE J W0 6= J 147

16q13 +16q12 +16q11 +15q10 +13q9 +12q8 +10q7 +8q6 +6q5 +5q4 +3q3 +2q2 +q +1 k = q19(q6 + q5 + 2q4 + 2q3 + 2q2 + q + 1) 148APPENDIX A. PARAMETERS OF GENERALIZED KNESER GRAPHS Bibliography

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[31] G. O. H. Katona. A simple proof of the Erd¨os-ChaoKo-Rado theorem. Journal of Combinatorial Theory, Series B, 13(2):183–184, 1972.

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[33] D. J. Kleitman. On an extremal property of antichains in partial orders. The LYM property and some of its implications and applications. In Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974), Part 2: Graph theory; foundations, partitions and combinatorial geometry, pages 77–90. Math. Centre Tracts, No. 56. Math. Centrum, Amsterdam, 1974.

[34] M. Kneser. Aufgabe 300. Jahresbericht der Deutschen Mathematiker- Vereinigung, 58:27, 1955. 152 BIBLIOGRAPHY

[35] L. Lov´asz.Flats in matroids and geometric graphs. In P. J. Cameron, editor, Combinatorial surveys, Proc. 6th British Comb. Conf., Egham, pages 45–86. Acad. Press, London, 1977.

[36] L. Lov´asz. Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory, Series A, 25:319–324, 1978.

[37] D. Lubell. A short proof of Sperner’s lemma. Journal of Combinatorial Theory, 1:299, 1966.

[38] J. Matouˇsek.A combinatorial proof of Kneser’s conjecture. Combina- torica, 24(1):163–170, 2004.

[39] L. D. Meshalkin. Generalization of Sperner’s theorem on the number of subsets of a ﬁnite set. Theory of Probability and it’s Applications, 8:203–204, 1963.

[40] K. Metsch. Blocking sets in projective spaces and polar spaces. J. Geom., 76(1-2):216–232, 2003. Combinatorics, 2002 (Maratea).

[41] J. C. Meyer. Quelques probl`emesconcernant les cliques des hyper- graphes h-complets et q-parti h-complets. In Hypergraph Seminar (Proc. First Working Sem., Ohio State Univ., Columbus, Ohio, 1972; dedicated to Arnold Ross), pages 127–139. Lecture Notes in Math., Vol. 411. Springer, Berlin, 1974.

[42] K. S. Sarkaria. A generalized Kneser conjecture. Journal of Combina- torial Theory, Series B, 49:236–240, 1990.

[43] E. R. Scheinerman and D. H. Ullman. Fractional Graph Theory. Wiley- Interscience, 1997.

[44] E. Sperner. Ein Satz ¨uber Untermengen einer endlichen Menge. Math- ematische Zeitschrift, 27(1):544–548, 1928.

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antichain, 8 Desargues configuration, 21 diagonal criterion, 34 Bárány, I., 66 dimension basis, 20 projective dimension, 20 binomial coefficient, 8 distance, 10 q-binomial coefficient, 28 division ring, 16 blocking set, 95 Dol’nikov, V.L., 68 minimal, 95 BN-pair, 99 edge, 9 Borel subgroup, 100 join, 9 Borsuk, K., 66 Eisfeld, J., 72 Bose, R.C., 71 EKR family, 31 Burton, R.C., 71 maximal, 31 embedding, 14 Cartan subgroup, 100 Erd˝os-Ko-Radotheorem, 31 chain, 8 extension map, 62 Chowdhury, A., 69 family (of sets), 8 chromatic number, 11 Fekete, M., 13 fractional chromatic number, field, 16 12 characteristic, 17 multiple chromatic number, 12 finite field, 16 clique, 10 order, 16 clique number, 11 Frankl, P., 33 coclique, 10 chamber-type coclique, 80 Gale, D., 66 connected component, 10 Gaussian coefficient, 24 Coxeter diagram, 101 generalized quadrangle, 94 Coxeter group, 101 geometry, see incidence structure Coxeter matrix, 101 Godsil C., 69 Coxeter system, 101 Godsil, C., 33 cycle, 10 graph, 9

154 INDEX 155

coloring, 11 LYM inequality, 29 k-fold coloring, 12 minimal coloring, 11 maximal arc, 85 minimal multiple coloring, 12 maximal intersecting family, 31 complement, 10 maximal split torus, 100 complete graph, 10 Meshalkin, L.D., 29 connected, 10 Milner, E.C., 36 edgeless graph, 11 multiplication map, 62 empty graph, 10 neighbor, 9 homomorphism, 14 Newman, M.W., 33 regular, 9 subgraph, 10 ovoid, 96 induced subgraph, 10 tensor product, 116 Pappus conﬁguration, 21 group, 15 parabolic subgroup, 105 commutative, 16 partial order, 9 of Lie type, 100 partially ordered set, 9 order, 16 path, 10 simple path, 10 Hilton, A.J.W., 36 point, 17 Hsieh, W.H., 32 point pencil, 31 hyperplane, 20 center, 31 incidence relation, 17 point-line geometry, see incidence incidence structure, 17 structure independence number, 11 point-line incidence structure, see independent set, 10 incidence structure isomorphism, 14 polar disjointness graph, 87 polar space, 25 Katona, G., 31 generator, 25 Kneser, E. poset, 9 q-Kneser graph, 61 projective line, 18 Kneser graph, 15, 59 projective plane, 18 Polar q-Kneser graph, 88 projective space, 18 Kneser, M., 66 Desarguesian projective space, 21 line, 17 Pappian projective space, 21 linearly independent set, 19 loop, 116 rank, 20 Lov´asz,L., 66 root system, 101 Lubell, D., 29 Royle G., 69 156 INDEX

Sarkaria, K.S., 68 set system, 8 span, 19 Sperner, E., 29 Sperner family, 8 Sperner’s theorem, 28 spread, 91 Stahl, S., 62 Storme, L., 72 subﬁeld, 17 subspace, 19 Sziklai, P., 72

Tits system, 100 Tucker, A.W., 68 vertex, 9 adjacent, 9 coloring, 11 degree, 9 joined, 10

Weyl group, 100 Wilson, R.M., 33

Yamamoto, K., 29 Samenvatting

Dit proefschrift behandelt de kruisbestuiving van extremale combinatoriek en eindige meetkunde. Combinatoriek beschrijft discrete (en doorgaans eindige) objecten. Extremale combinatoriek bestudeert hoe groot of hoe klein een collectie eindige objecten kan zijn onder bepaalde voorwaarden. Deze objecten kunnen verzamelingen, grafen, vectoren en dergelijke meer zijn. De vragen in dit gebied komen vaak uit de informatietheorie en comput- erwetenschap. Een andere tak van de combinatoriek is eindige meetkunde over lichamen van orde q. Sommige deelstructuren in een eindige meetkunde kunnen beschouwd worden als versies van projectieve vlakken voor q = 1, zelfs al bestaan er geen lichamen van orde 1. Een driehoek in een projectief vlak is een goed voorbeeld van dit fenomeen.

Na een inleidend hoofdstuk geeft Hoofdstuk 2 een overzicht van een aantal klassieke problemen in de extremale combinatoriek en hun q-analoga. De meest recente resultaten voor de q-analoga van de stelling van Sperner, de Erd˝os-Ko-Radostelling en verschillende versies van de stelling van Bol- lob´asworden beschreven. We tonen een aantal nieuwe resultaten voor het q-analogon van de Hilton-Milner stelling. Verder beschrijven we een nieuwe grens voor de minimale grootte van het q-analogon van kleine maximale klieken. We besluiten dat, in sommige gevallen, resultaten voor een q-analogon kunnen gevonden worden door gebruik te maken van dezelfde methoden als in het origineel geval. In sommige gevallen is het antwoord zelfs identiek als in het origineel probleem. In andere gevallen kunnen de techieken die gebruikt worden voor het q-analogon ook gebruikt worden om grenzen in het origineel probleem te verbeteren.

Het derde hoofdstuk beschrijft de reeds gekende resultaten voor het kleuringsgetal van de Knesergraaf en geeft nieuwe grenzen voor het kleuringsgetal van het q-analogon. De knopen van een Knesergraaf zijn deelverza- melingen (van een vaste grootte) van een gekozen verzameling. Twee knopen

157 158 INDEX zijn verbonden als ze disjunct zijn in hun voorstelling als deelverzameling. M. Kneser formuleerde in 1955 een vermoeden over het kleuringsgetal van deze grafen. Dit vermoeden werd in 1987 bewezen door L. Lóvasz. Het kleuringsgetal van twee kleine gevallen in het q-analogon werd gevonden door J. Eisfeld, L. Storme en P. Sziklai in 2001 en door A. Chowdhury, C. Godsil en G. Royle in 2006. We beschrijven een asymptotisch resultaat voor alle gevallen (behalve voor éénparameterfamilie) dat gebruik maakt van het q-analogon van de Hilton-Milner stelling.

Hoofdstukken 4, 5 en 6 beschrijven andere q-analoga van de Kneser grafen. In Hoofdstuk 4 deﬁni¨erenwe een familie Kneser grafen over paren van incidente punten en hypervlakken van een projectieve ruimte. We beschrijven grote maximale coklieken in deze grafen en bewijzen dat deze coklieken de grootst mogelijke zijn in kleine dimensies. We formuleren het vermoeen dat dit het geval is in alle dimensies. In Hoofdstuk 5 veralgemenen we de Kneser grafen over eindige polaire ruimten in plaats van projectieve ruimten. We geven beschrijvingen en kleuringsgetallen van deze grafen in een aantal gevallen. Hoofdstuk 6 beschrijft de veralgemening van Kneser grafen over nevenklassen ten opzichte van parabolische subgroepen van Chevalley groepen (geparametriseerd door q). We beschrijven deze grafen en bepalen kleuringsgetallen in een aantal gevallen waar q = 1. Vervolgens beschouwen we deze grafen voor algemene q en bekijken we welke resultaten in het q = 1 geval kunnen vertaald worden naar het algemene geval.

We vonden in dit proefschrift verschillende mogelijkheden voor het ver- band tussen de grenzen voor het q = 1 geval en het algemene geval. In een aantal voorbeelden zijn de grenzen voor het algemene geval identiek aan deze in het q = 1 geval. In andere voorbeelden bekomen we de grenzen voor het q = 1 geval door de limiet voor q → 1 te nemen in de grenzen in het algemeen geval. In sommige voorbeelden tenslotte zijn de resultaten in het q = 1 geval totaal verschillend van deze in het algemeen geval. Summary

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon.

Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner’s theorem, the Erd˝os-Ko- Rado theorem and several versions of Bollob´as’stheorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem.

The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a ﬁxed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. L´ovasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme

159 160 INDEX and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem.

Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case.

In this thesis we found diﬀerent possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q → 1 in the general bound. Other cases show a completely diﬀerent bound in both cases. Acknowledgements

First of all, I would like to thank my co-supervisor Aart Blokhuis. It was Aart who introduced me to the beautiful cross-fertilization between extremal combinatorics and ﬁnite geometry. Furthermore this thesis would not have been here without the help of my supervisors Andries Brouwer and Arjeh Cohen. Andries continually came up with new ideas and improvements and Arjeh made sure the research was kept on track and helped me with the results in the last chapter of this thesis.

I would also like to thank Tam´asSz˝onyi, who made a great contribution to the Hilton-Milner results in Chapter 2. The other members of my reading committee are Hans Cuypers and Leo Storme. Their work in reading my thesis and commenting on it is greatly appreciated. I also owe a great deal to the Incidence Geometry research group in Gent, where I completed my Master thesis. Especially I would like to thank Frank De Clerk who introduced me to Finite Geometry, Hendrik Van Maldeghem who introduced me to the Discrete Algebra group in Eindhoven and Jan De Beule, with whom I had a lot of interesting discussions.

There are a lot of people who made the four years in Eindhoven very pleasant. First of all, thanks to the other Ph.D. students of the Discrete Algebra group and Coding and Crypto group, lunches were never boring. Furthermore, Andrey was a great room mate during those years, always ready to have some fun or discuss something serious. I’m very grateful I met Ellen during my ﬁrst month in Eindhoven. She became a good friend who was always there to have a good talk or share some good advice with me.

Finally I would like to thank my family. My parents, grandparents, and sister Isabelle for their support, encouragement and understanding. And last but not least my wife Inge for her endless love.

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Curriculum Vitae

Tim Mussche was born on October 12, 1981 in Eeklo, Belgium. Between 1993 and 1999 he was a high school student at the Sint Vincentius college in Eeklo.

In 1999 he started studying at the University of Ghent where he took a variety of courses in mathematics, physics, computer science and philoso- phy. He received his bachelor’s degree in mathematics in 2001. Tim wrote his Master’s thesis on model theory and generalized quadrangles under the supervision of prof.dr. Hendrik Van Maldeghem. In 2003 he received his Master’s degree in pure mathematics.

Thanks to professor Van Maldeghem Tim came in contact with the Dis- crete Algebra and Geometry group at the Technical University of Eind- hoven. In 2003 he joined this group where he was a Ph.D student under the supervision of prof.dr. Andries Brouwer, prof.dr. Arjeh Cohen and dr. Aart Blokhuis. During this time he researched the connections between extremal combinatorics and ﬁnite geometry in the context of generalized Kneser graphs. This thesis is the result of this research.

Tim is currently employed in Belgium as business process management consultant at the consultancy ﬁrm MOBIUS.¨

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