Problems in Positional Games and Extremal Combinatorics Christopher Kusch Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften am Fachbereich f¨urMathematik und Informatik der Freien Universit¨atBerlin Berlin, 2017 ii Erstgutachter: Prof. Tibor Szab´o,PhD (Betreuer) Zweitgutachter: Prof. Dr. Anusch Taraz Tag der Disputation: 4.10.2017 Contents Preface ix Acknowledgement xi 1 Introduction1 1.1 Strong Ramsey games...............................2 1.1.1 Introduction to strong games.......................2 1.1.2 The strong Ramsey game.........................3 1.1.3 The transition to the infinite board....................3 1.2 Maker-Breaker G-games and van der Waerden games..............5 1.2.1 Introduction to Maker-Breaker games..................5 1.2.2 The Maker-Breaker G-game........................7 1.2.3 The van der Waerden game and its generalisation............9 1.2.4 General winning criteria.......................... 11 1.3 Hypergraphs with Property O ........................... 13 1.3.1 Introduction of the problem........................ 13 1.3.2 Results: an improved upper bound.................... 15 1.4 Shattering extremal families............................ 15 1.4.1 Shattering.................................. 15 1.4.2 The elimination conjecture........................ 18 1.4.3 The results................................. 18 2 Strong Ramsey games: Drawing on an infinite board 21 2.1 Overview of the proof............................... 21 2.2 Sufficient conditions for a draw.......................... 22 2.2.1 No threat.................................. 24 2.2.2 Special threat................................ 25 2.2.3 Standard threats.............................. 27 iii iv CONTENTS 2.3 An explicit construction.............................. 29 2.4 Concluding remarks and open problems..................... 37 3 General Winning Criteria for Maker-Breaker games 39 3.1 The threshold bias of the k-AP game....................... 39 3.1.1 Maker's strategy - a deterministic approach............... 40 3.1.2 Maker's strategy - a probabilistic approach............... 40 3.1.3 Breaker's strategy............................. 42 3.2 Proof of Theorem 1.2.8 { Maker's Strategy................... 45 3.2.1 Proof of Theorem 1.2.8 from Theorem 3.2.3............... 45 3.2.2 Proof of Theorem 3.2.3.......................... 47 3.3 Proof of Theorem 1.2.9 { Breaker's Strategy................... 48 3.3.1 Preliminaries and Definitions....................... 49 3.3.2 Two important strategies for Breaker.................. 51 3.3.3 Proof of Theorem 3.3.1.......................... 53 3.4 Proof of Theorem 1.2.2 - Maker-Breaker G-game................ 54 3.5 Generalising the van der Waerden game..................... 56 3.5.1 Proper solutions.............................. 57 3.5.2 Solutions with repeated entries...................... 60 3.6 Proof of Theorem 3.5.5 { Generalised van der Waerden games......... 62 3.6.1 Preliminaries for Linear Systems..................... 63 3.6.2 Proof of Theorem 3.5.5.......................... 69 3.7 Concluding remarks and open problems..................... 72 3.7.1 Obtaining constants............................ 72 3.7.2 A corollary for strictly balanced structures............... 73 3.7.3 A remark about repeated entries..................... 74 3.7.4 The probabilistic intuition......................... 74 3.8 Appendix to Chapter 3............................... 76 4 Hypergraphs with Property O 79 4.1 Proof of the upper bound............................. 79 4.2 3-uniform hypergraphs on 6 vertices having property O............ 82 4.3 Concluding Remarks and open problems..................... 83 5 Shattering extremal families 85 5.1 An approach to the elimination conjecture.................... 86 5.1.1 Analysing the equation jH(S)j = jF(S; h)j ................ 87 CONTENTS v 5.1.2 Outline of the new approach....................... 87 5.1.3 Proof of Theorem 1.4.9.......................... 88 5.2 Small Sperner systems............................... 90 5.2.1 An equivalent conjecture.......................... 90 5.2.2 Proof of Theorem 1.4.10.......................... 92 5.3 Gr¨obnerBases................................... 93 5.3.1 A short introduction to Gr¨obnerbases.................. 94 5.3.2 Gr¨obnerbases and s-extremal families.................. 95 5.4 Concluding remarks and open problems..................... 97 5.5 Appendix to Chapter 5............................... 99 Zusammenfassung 101 Eidesstattliche Erkl¨arung 103 Bibliography 104 vi CONTENTS List of Figures 2.1 The 5-uniform hypergraph H. The black line from v1 to v9 represents the tight path consisting of the edges e1; : : : ; e5....................... 30 4.1 The 3-graph H with the edge (a1; a2; b2) depicted................ 82 vii viii LIST OF FIGURES Preface This thesis consists of five Chapters. The main purpose of the first chapter is to serve as an introduction. We will define all necessary concepts and discuss the problems that are studied in this thesis as well as stating the main results of it. This thesis deals with two different directions of extremal graph theory. In the first part we consider two kinds of positional games, the so-called strong games and the Maker-Breaker games. In the second chapter we consider the following strong Ramsey game: Two players take turns in claiming a previously unclaimed hyperedge of the complete k-uniform hypergraph on n vertices until all edges have been claimed. The first player to build a copy of a predetermined k-uniform hypegraph is declared the winner of the game. If none of the players win, then the game ends in a draw. The well-known strategy stealing argument shows that the second player cannot expect to ever win this game. Moreover, for sufficiently large n, it follows from Ramsey's Theorem for hypergraphs that the game cannot end in a draw and is thus a first player win. Now suppose the game is played on the infinite k-uniform complete hypergraph. Strategy stealing and Ramsey's Theorem still hold and so we might ask the following question: is this game still a first player win or a draw. In this chapter we construct a 5-uniform hypergraph for which the corresponding game is a draw. This chapter is based on [45]. In the third chapter we consider biased (1 : q) Maker-Breaker games: Two players called Maker and Breaker alternate in occupying previously unoccupied vertices of a given hyper- graph H. Maker occupies 1 vertex per round and Breaker occupies q vertices. Maker wins if he fully occupies a hyperedge of H and Breaker wins otherwise. One of the central questions in this area is to find (or at least approximate) the maximal value of q that allows Maker to win the game. In this chapter we prove two new general winning criteria -one for Maker and one for Breaker- and apply them to two types of games. In the first type, the target is a fixed uniform hypergraph and in the second it is a solution to an arbitrary but fixed linear system of inhomogeneous equations. This chapter is based on [53]. ix x PREFACE The second part of this thesis deals with two types of questions from extremal combi- natorics. The first type is of the following form: suppose we are given a certain property of hypergraphs, what is the minimum possible number of edges a k-uniform hypergraph can have such that it does have the property of interest. The second type goes in the different direction. If a certain family of hypergraphs has the minimum number of edges satisfying a certain property, i.e. if the family is extremal in that sense, then how can we characterise it and what operations can we perform while maintaining extremality? In the fourth chapter, we investigate the minimum number f(k) of edges a k-uniform hypergaph having Property O can have. In [24] it is shown that k! ≤ f(k) ≤ (k2 ln k)k!, where the upper bound holds for sufficiently large k. We improve the upper bound by a k k factor of k ln k showing f(k) ≤ b 2 c + 1 k! − b 2 c(k − 1)! for every k ≥ 3. We also answer a question regarding the minimum number n(k) of vertices a k-uniform hypergaph having Property O can have. This chapter is based on [51]. In the fifth chapter, we consider shattering extremal set systems. A set system F ⊆ 2[n] is said to shatter a given set S ⊆ [n] if 2S = fF \ S : F 2 Fg. The Sauer-Shelah Lemma states that in general, a set system F shatters at least jFj sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly jFj sets. The so-called elimination conjecture, independently formulated by M´esz´arosand R´onyai as well as by Kuzmin and Warmuth, states that if a family is shattering-extremal then one can delete a set from it and the resulting family is still shattering-extremal. We prove this conjecture for a class of set systems defined from Sperner systems and for Sperner systems of size at most 4. Furthermore we continue the investigation of the connection between shattering extremal set systems and Gr¨obnerbases. This chapter is based on the extended abstract [52]. Notation Throughout this thesis we will use fairly standard notation, generally follow- ing [23]. For the convenience of the reader, we have included the required definitions needed to understand a particular chapter in the corresponding sections of the introduction. In chapter 4 we make use of asymptotic notation. Given two functions f; g : N ! R we write f = O(g) if there exists a constant C > 0 such that for n sufficiently large jf(n)j ≤ Cjg(n)j holds. We write f = Ω(g) if g = O(f), and f = θ(g) if f = O(g) and f = Ω(g). We write f(n) f = o(g) if g(n) ! 0 as n tends to infinity. If g = o(f) then we write f = !(g). Finally, if f(n) g(n) ! 1 as n tends to infinity, we write f ∼ g and say that f and g are asymptotically equal.