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 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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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 theo- retic 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 general- ize 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 2 X, • Antisymmetry: if x y and y x then x = y for all x; y 2 X, and • Transitivity: if x y and y z then x z for all x; y; z 2 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 compa- rable. 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.