Extremal Combinatorics in Generalized Kneser Graphs

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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 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. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. 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 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.
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