GAMES, GRAPHS, AND GEOMETRY BY WESLEY PEGDEN A dissertation submitted to the Graduate School|New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of J´ozsefBeck and approved by New Brunswick, New Jersey May, 2010 ABSTRACT OF THE DISSERTATION Games, Graphs, and Geometry by Wesley Pegden Dissertation Director: J´ozsefBeck This thesis concerns four separate topics: the balanced counterpart of the Hales-Jewett number, the maximal density of k-critical triangle-free graphs, Euclidean sets resilient to an `erosion' operation, and an extension of the Local Lemma which can be applied in a game setting. For the Hales-Jewett number, our motivation comes from a desire to show that there are infinitely many `delicate' Tic-Tac-Toe games. Roughly speaking, these are games where neither player has a simple reason for having a winning/drawing strategy. The first part of this thesis concerns the translation of bounds on the famous `Hales-Jewett number' into bounds on the `Halving Hales-Jewett number', its `balanced' version, which give the desired game-theoretic consequences. The second part of this thesis concerns k-critical triangle-free graphs: can they have quadratic edge-density, independent of k as k grows large? This question has close connections both to the study of the density of critical graphs, and the study of the chromatic number of triangle-free graphs. Surprisingly, we are able to determine the exact asymptotic density of k-critical triangle-free graphs for k 6, and even for ≥ pentagon-and-triangle-free graphs. In the third part, we will consider a simple erosion operation on sets in Euclidean space, which roughly represents the operation of `shaving off' points near the boundary ii of a set. We will give a complete characterization of sets whose shape is unchanged by this operation. Finally, in the fourth part, we will generalize the classical Lov´aszLocal Lemma to a `Lefthanded' version, which, roughly speaking, allows one to ignore dependencies `to the right' when making an application of the Local Lemma to bad events which have an underlying order. This will allow us to prove game-theoretic analogs of classical results on nonrepetitive sequences, representing the first successful applications ofa Local Lemma to games. iii Acknowledgements I would like to thank Andr´asGy´arf´asfor helpful discussions and encouragement on the problem considered in Chapter 3 (and in particular, for giving the construction illustrated in Figure 3.3). For Chapter 5, thanks are due for helpful conversations with Jaros law Grytczuk regarding nonrepetitive sequences. I would also like to thank Jeff Kahn, Joel Spencer, and Doron Zeilberger for their service on my committee, and Mike Saks for discussions and advice in his role as director of the Graduate Program. Finally, I thank my advisor, J´ozsefBeck, for his constant and enthusiastic encouragement and support, and for many, many discussions on all the problems considered in this thesis. iv Table of Contents Abstract :::::::::::::::::::::::::::::::::::::::: ii Acknowledgements ::::::::::::::::::::::::::::::::: iv 1. Introduction ::::::::::::::::::::::::::::::::::: 1 2. Tic-Tac-Toe and the halving Hales-Jewett number ::::::::::: 12 2.1. Which Halving Hales-Jewett number? . 14 2.2. Proving HJ 1 (n) HJ(n 2) ........................ 15 2 ≥ − 2.3. Further questions . 19 3. Odd-girth in dense k-critical graphs :::::::::::::::::::: 21 3.1. Avoiding triangles (constructions) . 23 3.1.1. Constructing Gk ........................... 27 3.1.2. Density of G (for k 6) ...................... 28 k ≥ 3.1.3. Density of G5 ............................. 29 3.2. Avoiding pentagons (existence result) . 30 3.3. Avoiding more odd cycles (` 7) for k =4 ................ 34 ≥ 3.4. Further Questions . 35 4. Sets resilient to erosion :::::::::::::::::::::::::::: 39 4.1. The bounded case . 42 4.2. The general convex case . 49 3 4.2.1. Unbounded convex examples in R ................. 50 4.2.2. Characterizing all resilient convex bodies in Rn .......... 51 4.2.3. Convex sets resilient to expansion . 55 4.3. The nonconvex case . 56 v 4.3.1. Characterizing nonconvex resilient sets . 58 4.3.2. Fractals and erosion . 62 4.4. Further Questions . 64 5. Winning strategies from a Lefthanded Local Lemma :::::::::: 67 5.1. An easier game . 72 5.2. Lefthanded Local Lemma . 76 5.3. Thue-type binary sequence games . 79 5.3.1. Long identical intervals can be made far apart . 79 5.3.2. Adjacent intervals can be made very different . 81 5.4. c-ary nonrepetitive sequence games . 83 5.5. Pattern avoidance . 90 5.6. Further Questions . 92 References ::::::::::::::::::::::::::::::::::::::: 97 Vita ::::::::::::::::::::::::::::::::::::::::::: 101 vi 1 Chapter 1 Introduction Discrete Mathematics is a young branch of Mathematics in all important senses. Its grand classical problems (the 4-coloring theorem, for example, or the perfect graph conjectures) have been solved in recent memory, rather than in the distant past, while defining problems with no solution in sight (such as the P=NP? problem) aredecades rather than centuries old. At the same time, completely new and surprising directions of inquiry are constantly being discovered, as is the case, for example, with the relatively recent attention focused on combinatorial games, or the rise of additive combinatorics as a major focus of attention. Discrete mathematics is a discipline which shows no shortage of new directions; no shortage of new mysteries to be uncovered. In a reflection of the varied richness of the field, this thesis is not confined totheex- amination of a single problem, but instead concerns several problems we have addressed; the only common thread is the pursuit of nice questions. We will use new results about the classical Hales-Jewett number to show the existence of infinitely many `delicate' Tic-Tac-Toe games, achieve uncharacteristically optimal results through a new inquiry in the classical area of color-critical graphs, prove a surprising characterization of Eu- clidean sets `resilient' to `erosion', and develop a generalization of the Lov´aszLocal Lemma which allows a new kind of probabilistic approach to certain combinatorial games. *** Motivation for the study of the Hales-Jewett number comes from nd Tic-Tac-Toe games, higher-dimensional analogs of the 32 (or 3 3) game played by children. The classical × Hales-Jewett number HJ(n) can be defined as the smallest dimension D for which it is impossible to mark the cells of the nD hypercube with x's and o's in such a way that 2 there are no `n-in-a-line' Tic-Tac-Toe winning sets which are either all x's or all o's. This immediately implies, then, that whenever d HJ(n), a Tic-Tac-Toe game on the ≥ nd board cannot end in a draw. The simple-yet-powerful strategy stealing argument implies that Player 2 can never win a Tic-Tac-Toe game when Player 1 is playing perfectly, and so Player 1 has a winning strategy at Tic-Tac-Toe on the nd board when d HJ(n)|for these games, the Ramsey-theoretic Hales-Jewett number is enough to ≥ deduce the existence of a winning strategy for Player 1. To this day, upper bounds on the Hales-Jewett number are the only results which prove the existence of winning strategies for large Tic-Tac-Toe games. Nevertheless, it is perhaps worth noticing that Tic-Tac-Toe games where d HJ(n) ≥ are a bit strange from the standpoint of competitive play. After all, these are games where a draw is impossible, and the players are simply competing to be the first to get n-in-a-line. This kind of `winning by Ramsey theory' seems quite different from the behavior of more familiar Tic-Tac-Toe games. The 43 Tic-Tac-Toe game (Qubic) was played competitively, and was shown by Patashnik's huge computer-assisted work [44] to be a first player win, but this is a game where drawing positions are plentiful: Player 1 wins because he is able to skillfully avoid them. This is what is called a `delicate win' for Player 1|he can win the game even though a draw is possible. Normally, `winning' in Tic-Tac-Toe means getting n-in-a-line before the other player. It turns out that this is much harder than trying to get n-in-a-line if you are willing to let your opponent beat you to it. Although anyone who has played it knows that ordinary 3 3 Tic-Tac-Toe ends in a draw so long as Player 2 makes no mistakes, it is actually × easy for Player 1 to get 3 x's in a row in this game|he just may have to let Player 2 get 3 o's in a row first. This alternative goal corresponds to what is called a `weak win' in a positional game, where Player 1 achieves his goal, although not necessarily before Player 2. The `fake probabilistic theory' developed by Beck allows analysis of `weak wins' for positional games, and in particular, has succeeded at determining the behavior of the `weak win' with respect to the dimension for Tic-Tac-Toe (in stark contrast to the lack of knowledge about the growth of the Hales-Jewett number, for example). In particular, Beck has shown [5] that Player 1 has a strategy for a `weak win' in the nd 3 log 2 2 Tic-Tac-Toe game if d > 2 n , while, on the other hand, Player 2 can prevent Player 2 1 from even achieving a weak win if d < ( log 2 o(1)) n . When Player 2 can prevent 16 − log n Player 1 from achieving a weak win, we say he has a `strong draw'.