GAMES, GRAPHS, AND GEOMETRY

Transcription

1 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 József Beck and approved by New Brunswick, New Jersey May, 010

2 ABSTRACT OF THE DISSERTATION Games, Graphs, and Geometry by Wesley Pegden Dissertation Director: József Beck 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

3 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 Lovász Local 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 of a Local Lemma to games. iii

4 Acknowledgements I would like to thank András Gyárfás for 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 Jaroslaw 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, József Beck, for his constant and enthusiastic encouragement and support, and for many, many discussions on all the problems considered in this thesis. iv

5 Table of Contents Abstract ii Acknowledgements iv 1. Introduction Tic-Tac-Toe and the halving Hales-Jewett number Which Halving Hales-Jewett number? Proving HJ 1 (n) HJ(n ) Further questions Odd-girth in dense k-critical graphs Avoiding triangles (constructions) Constructing G k Density of G k (for k 6) Density of G Avoiding pentagons (existence result) Avoiding more odd cycles (l 7) for k = Further Questions Sets resilient to erosion The bounded case The general convex case Unbounded convex examples in R Characterizing all resilient convex bodies in R n Convex sets resilient to expansion The nonconvex case v

6 Characterizing nonconvex resilient sets Fractals and erosion Further Questions Winning strategies from a Lefthanded Local Lemma An easier game Lefthanded Local Lemma Thue-type binary sequence games Long identical intervals can be made far apart Adjacent intervals can be made very different c-ary nonrepetitive sequence games Pattern avoidance Further Questions References Vita vi

7 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) are decades 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 to the examination 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 Euclidean sets resilient to erosion, and develop a generalization of the Lovász Local Lemma which allows a new kind of probabilistic approach to certain combinatorial games. * * * Motivation for the study of the Hales-Jewett number comes from n d Tic-Tac-Toe games, higher-dimensional analogs of the 3 (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 n D hypercube with x s and o s in such a way that

8 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 n d board cannot end in a draw. The simple-yet-powerful strategy stealing argument implies that Player 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 n d 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 4 3 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 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 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. 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 n d

9 3 Tic-Tac-Toe game if d > log n, while, on the other hand, Player can prevent Player 1 from even achieving a weak win if d < ( log 16 n o(1)) log n. When Player can prevent Player 1 from achieving a weak win, we say he has a strong draw. In cases like the familiar 3 3 Tic-Tac-Toe game where this is not the case we say that Player has a delicate draw. Such cases present what are perhaps the most interesting drawing strategies for Player, since they depend on Player s ability to create his own threats to prevent Player 1 from winning. It is natural to wonder how many Tic-Tac-Toe games fall into the classes of delicate win and delicate draw described above, since these seem to be the most interesting classes of Tic-Tac-Toe games, in some sense (and, certainly, the most difficult classes to analyze mathematically). Unfortunately, other than the 3 and 4 3 games already mentioned, no other examples of delicate Tic-Tac-Toe games are known at all. And it is unknown, for example, whether either class could consist of just the single game already known in it. On the other hand, through recent work with Beck and Vijay, we have shown that the union of these two classes must contain infinitely many Tic-Tac-Toe games. particular, our joint paper [7] contains a new exponential lower bound HJ(n) n/4 3n 4 In on the Hales-Jewett number HJ(n), improving the previous best-known bound HJ(n) n from the original paper of Hales and Jewett. This almost implies that there are terminal drawing positions in n d Tic-Tac-Toe games when d < n/4 3n 4, but not quite, since the colorings guaranteed to exist for this result are not necessarily balanced colorings (the number of x s and o s may differ by more than one). The quantity we would like to bound for implications to Tic-Tac-Toe is the halving Hales-Jewett number HJ 1 (n), defined as the minimum d 0 for which any balanced coloring of n d results in an all-x s or all-o s n-a-line when d d 0. Coupled with Beck s results on the weak-win, a superquadratic bound on HJ 1 (n) would imply that there are infinitely many Tic-Tac-Toe games where d is large enough relative to n that Player doesn t have a strong draw, but where d is small enough relative to n that Player 1 doesn t have a strong win either.

10 4 In Chapter we will present a proof (also in [7]) that HJ 1 (n) HJ(n ), implying an exponential lower bound for HJ 1 (n) from the already mentioned bound for HJ(n), and implying the existence of infinitely many delicate Tic-Tac-Toe games. * * * There is a recurring theme in graph theory concerning the comparison of bipartite graphs with triangle-free graphs, since the former are graphs without any odd cycles, and the latter are graphs avoiding just the shortest kind of odd cycles. To what extent does avoiding triangles force these families behave similarly? It is an old result that triangle-free graphs can have arbitrarily large chromatic number, but the classical constructions to demonstrate this (due to Zykov [56] and Mycielski [40], for example) are strikingly sparse graphs. This suggests that perhaps triangle-free graphs have trouble distinguishing themselves from bipartite graphs when they are required to have lots of edges (a suggestion further encouraged by the fact that a triangle-free graph with the maximum possible number of edges is a complete bipartite graph). We need to be careful what kind of questions we ask, however. In fact, it is easy to see that there are triangle-free graphs of large chromatic number and with a quadratic number of edges, since we can take the disjoint union of a k-chromatic triangle-free graph with a large complete bipartite graph. These kinds of considerations motivated Erdős and Simonovits to ask about the minimum degree of triangle-free graphs with large chromatic number, an issue which has recently been essentially resolved [11]. One can use the concept of k-criticality to ask another kind of question about dense triangle-free graphs. A graph is k-critical if it is k-chromatic and removing any edge results in a (k 1)-critical graph, thus it is clear that if we create dense trianglefree graph by taking disjoint unions with bipartite graphs, the results will be far from critical. This motivates the question: does there exist a family of triangle-free critical graphs of arbitrarily large chromatic number, and each with > cn

11 5 edges for some fixed c > 0? 1 In Chapter 3 we will answer this question in the affirmative, by proving the following theorem: Theorem 1.1. For k 4, there are triangle-free k-critical graphs with > (c k o(1))n edges, where here c , c , and c k = 1 4 for all k 6. Apart from the connection to the study of triangle-free graphs, this issue is closely related to the study of the density of critical graphs, which began with a question of Erdős, leading to the construction by Dirac of k-critical graphs (k 6) with > 1 4 n edges. Toft subsequently constructed a 4-critical graph with > 1 16 n edges; curiously, Toft s graph is triangle-free. Applying Mycielski s operation to Toft s graph is a simple way to get k-critical triangle-free graphs with 1 16 ( 3 4 )k 4 n edges; thus, the graphs have quadratic density for each k this does not answer the question we are interested in, however, since the density constant 1 16 ( 3 4 )k 4 tends to 0 as k grows large. Theorem 1.1, on the other hand, implies that quadratic density can in fact be maintained as k grows large, since the constant c k does not tend to 0 as k goes to infinity. Apart from that, the constant c k = 1 4 given in Theorem 1.1 for the case k 6 is even best possible, since Turán s theorem implies that triangle-free graphs have 1 4 n edges. In this sense, Theorem 1.1 stands out as an exception in the study of critical graphs to the rule that optimal results are not generally attainable. The problem is that there are no good techniques known to prove upper bound results for the density of critical graphs in fact, it may be that essentially all known upper bound results for the density of critical graphs come from applications of Turán s theorem. (Observe, for example, that a k-critical graph with more than k edges cannot contain Kk as a subgraph.) With Theorem 1.1, we are simply lucky that the lower bounds we achieve line up with the trivial Turán theorem upper bound. Chapter 3 also contains the proof of the following theorem with the same striking feature: 1 This question is quite similar to problems discussed by Erdős in [], but this particular formulation appears to have been ignored all the more suprising since it gives rise to optimum results.

12 6 not resilient resilient Figure 1.1: Some erosions of bounded shapes in R. The area in gray is what is removed by an erosion operation. Theorem 1.. For k 4, there are pentagon-and-triangle-free k-critical graphs with > (c 5,k o(1))n edges, where c 5,4 1 36, c 5,5 3 35, and c 5,k = 1 4 for k 6. It seems, in fact, that these two theorems give us the only natural familes of graphs (triangle-free and pentagon-and-triangle-free graphs, respectively) where the asymptotic density of critical members are known. Finally, in Chapter 1.1 we will also present the following result for 4-critical graphs of larger odd-girth: Theorem 1.3. For each odd l, there are 4-critical graphs without odd cycles of length l with > ( 1 l+1 ) n edges. * * * Consider a subset X of Euclidean space. We can define a simple erosion operation e r (X) = X \ y XC B(r, y) which removes from X any points at distance < r of the complement of X. (B(r, y) denotes an open ball of radius r about a point y.) The effect of this operation on some familiar shapes is shown in Figure 1.1. This operation is natural enough that it has been studied from a practical standpoint, as a model of pebble erosion (see, e.g., [0]). However, there is a quite natural theoretical question which is immediately suggested by the operation for which sets (and radii) does the erosion operation produce a set which is equivalent to the original under a Euclidean similarity transformation? When X is similar to e r (X), X is said to be resilient to erosion by the radius r. Referring again to Figure 1.1, it is clear that a closed disk is resilient to erosion, as are, for example, the closed bodies of regular polygons. Bodies of triangles are also resilient: when their erosion consists of more than just a single point, it is the body of another triangle with the same angles, and so is similar to the original set. The body of an irregular rectangle, on the other hand, is not resilient, since the ratio of

13 7 (a) A portion of an unbounded (and scale-invariant) version of Koch s curve. (b) Part of an unbounded resilient set derived from the unbounded version of Koch s snowflake. The area removed by two successive erosion operations is shown in two different shades of gray. Note that erosion makes it bigger. Figure 1.: Getting a resilient set from the Koch curve. the side-lengths changes upon erosion and of course, it should be clear that typical shapes will not be resilient to erosion by any positive radius. In Chapter 4, we will prove the following characterization of bounded resilient sets: Theorem 1.4. A bounded set X R n is resilient to erosion by some radius r > 0 if and only if it is a closed convex set with an inscribed ball of radius > r. The definition of an inscribed ball will be given in Chapter 4, but for now, we will point out that it is not a coincidence that all the resilient polygons from Figure 1.1 have inscribed circles. Although Theorem 1.4 provides a complete and elegant characterization of bounded resilient sets, it requires a surprisingly technical proof when the dimension n is greater than, due to annoying possibilities which must be allowed for for example, that the set X is resilient to erosion, with corresponding similarity transformations which always include irrational rotations. We will see in Chapter 4 that it is possible to extend Theorem 1.4 to a result which covers unbounded convex sets. It might seem reasonable, perhaps, to expect that all resilient sets are convex, so that this would constitute a complete characterization of resilient subsets of Eucliean space. This, it turns out, is not even close to being true,

14 8 Figure 1.3: A cropped example of a fractal-like unbounded resilient set from R. The gray area is the region removed from the set upon an erosion by the smallest radius by which it is erosion-resilient. The corresponding similarity transformation is the homothety which fixes the center of the region shown and increasing distances by a factor of 7.

15 9 as there are extremely pathological examples of nonconvex unbounded resilient sets. Figure 1.3 shows an example of an unbounded resilient set all of whose connected components are bounded, and yet which is resilient to erosion by arbitrarily large radii. Figure 1.(b) shows an example of another unbounded nonconvex set resilient to erosion, this time produced from the familiar Koch curve fractal. Note that both of these sets get bigger upon erosion, in the sense that the corresponding similarity transformations are distance-increasing. (These two sets are also examples which are resilient to erosion only by a discrete set of radii.) Most surprising is the fact that it is possible to characterize even such pathological examples of resilient sets: Theorem 1.5. A set X R n satisfies e r (X) = σ(x) for some distance-increasing similarity transformation σ if and only if we have X = e r/(α 1) (W ) for some set W which is scale-invariant under σ. (A set X is scale-invariant if we have X = σ(x) for some non-isometric similarity transformation in other words, if X looks the same at multiple scales.) In the case of Figure 1.(b), for example, the scale-invariant set guaranteed to exist by Theorem 1.5 is the lower component of the complement of the Koch curve of Figure 1.(a). In Chapter 4, We will consider the connections between resilient sets and mathematical fractals such as the Koch snowflake and Sierpinśki s triangle suggested by Theorem 1.5. We will see in Chapter 4 that if e r (X) = σ(x) and X is not distance-increasing then X must be convex, thus Theorem 1.5, along with the results covering convex resilient sets, completes a total characterization of Euclidean sets resilient to erosion. * * * In 1906, Axel Thue constructed [5,53] a ternary sequence without any consecutive repeated subwords (e.g.,, 11 never appears in this sequence). This construction began the study of nonrepetitive sequences, a subject rich with diverse questions (many of which are discussed in the survey of Grytczuk [30]) concerning, for example, the possibility of constructing sequences in which identical blocks are not

16 10 only nonadjacent but far apart. Any question about nonrepetitive sequences has a natural game analog: suppose Players 1 and take turns choosing digits of an unending c-ary sequence (Player 1 chooses the first digit from the set {1,,..., c}, Player chooses the second digit from the same set, Player 1 chooses the third digit, and so on). What kind of nonrepetitiveness can Player 1 achieve in this game? Of course, through imitation, Player can make repetitions of single digits. It turns out, however, that when c is sufficiently large and Player 1 is playing well, that is the best Player can hope to achieve: Theorem 1.6. Player 1 has a strategy in the 37-ary sequence game which ensures that there will be no identical adjacent blocks of length. Theorem 1.6 thus gives a natural game-theoretic analog to Thue s old theorem on nonrepetitive sequences. The biggest surprise regarding Theorem 1.6 is that it is proved with a Local Lemma. Previous attempts to apply the Local Lemma to games have failed, as the unknown second player s strategy typically introduces a mess of dependencies which removes any possibility of demonstrating the kind of only local dependence criterion needed to apply the Local Lemma. (In a nutshell, how can we argue that events in a game are independent, when Player s moves that influence each event are allowed to depend on previous moves in the game?) We avoid this difficulty by proving a generalization of the Lovász Local Lemma which allows one to ignore dependencies in one direction when there is a natural ordering underlying the bad events we are trying to avoid. In the setting of sequence games, this Lefthanded Local Lemma (presented in Chapter 5) allows one to ignore the problematic dependencies on future events which would normally make an application of the Local Lemma totally impossible in the game setting. (Dependencies on past events can be dealt with in a more conventional way using the Lopsided Local Lemma proved by Erdős and Spencer.) The same result allows us to prove game-theoretic analogs (presented in Chapter 5) of essentially all the classical theorems on nonrepetitive sequences, and the version

17 11 of the Local Lemma we developed to attack these problems has recently been used by Grytczuk, Przybylo, and Zhu [31] to achieve near-optimum results for the Thue-choicenumber, the list-chromatic analog of Thue s original theorem of ternary nonrepetitive sequences.

18 1 Chapter Tic-Tac-Toe and the halving Hales-Jewett number The familiar game of Tic-Tac-Toe is played on a 3 3 board. Players take turns marking the squares of the board with either x s (for Player 1) or o s (for Player ) until one of the Player s has achieved 3-in-a-line of their symbol. Although this familiar game of Tic-Tac-Toe is rather simple, it lends itself very naturally to generalization. When played on an [n] d hypercube, winning sets consist of sequences (c (1) j, c () j,..., c (d) j ) (1 j n) of n points in the hypercube, such that for any fixed k, the coordinate sequence {c (k) j } n j=1 is either constant, the increasing sequence 1,,..., n, or the decreasing sequence n, n 1,..., 1. (At least one of the c (i) s must be a moving coordinate, so that the line doesn t consist of a single point). This just generalizes the 8 familiar lines from the 3 3 board. The work of this chapter is concerned with the existence of delicate Tic-Tac-Toe games. As discussed in the Introduction, these are games for which either Player 1 has a winning strategy in spite of the fact that terminal drawing positions exist, or in which Player has a drawing strategy in spite of the fact that Player 1 has a weak win. Since Beck has shown that Player 1 has a weak win so long as d > log n, a logical strategy for proving the existence of delicate Tic-Tac-Toe games is proving a good lower bound on the Hales-Jewett number. The joint paper [7] with J. Beck and S. Vijay contains the lower bound HJ(n) n/4 3n 4, (.1) improving the bound HJ(n) n from the original paper of Hales and Jewett [3] by connecting the Hales-Jewett number to a quadratric progression version of the Van

19 13 der Waerden number. Note that a lower bound on the Hales-Jewett number is not sufficient to demonstrate the existence of terminal drawing positions on Tic-Tac-Toe boards. When d < HJ(n) we are guaranteed the existence of -colorings of [n] d in which there are no monochromatic winning sets, but we don t know the existence of a balanced coloring (in which the two color classes differ in size by at most 1) with the same property, and these are the colorings which constitute the terminal drawing positions of Tic-Tac-Toe games. (Moreover, the nature of the proof of (.1) is such that knowledge of balanced colorings in the Van der Waerden setting does not translate into information about balanced hypercube colorings). In the following sections, we will demonstrate that lower bounds on HJ(n) can be translated into bounds on the Tic-Tac-Toe-relevant halving Hales-Jewett number, through the following simple inequality: Theorem.1. For all integers n >, we have HJ(n ) HJ 1 (n) HJ(n). (.) We will see in the following section that there are extremely basic questions about the Halving Hales-Jewett number that may be completely intractable, making it perhaps surprising that it is possible to pin down HJ 1 (n) so precisely in terms of HJ(n). (Note that the upper bound HJ 1 (n) HJ(n) is immediate from the definitions.) The basic proof technique comes from decomposing the [n] d hypercube into smaller hypercubes. Observe, for example, that if n is even, we can partition the [n] d hypercube into d copies of the [n/] d hypercube, and if the cube is colored so that each of the d subhypercubes exhibits a terminal drawing position in the [n/] d Tic-Tac-Toe game, then the whole [n] d cube must be colored with a terminal drawing position, since any winning set in the whole cube contains winning sets in the subhypercubes. This decomposition proves the (not very useful!!) inequality HJ(n) HJ( n ). To get the bound HJ 1 (n) HJ 1 (n ), we will decompose the [n] d hypercube into

20 14 Figure.1: A decomposition of the [7] hypercube into a [5] hypercube, four [5] 1 hypercubes, and four [5] 0 hypercubes. such that any monochromatic line in the whole [7] hypercube would be monochromatic on one of the cubes of dimension 1 in the decomposition. the central [n ] d hypercube, along with the d different [n ] d 1 central subhypercubes of its faces, and then the d(d 1) different [n ] d central subhypercubes of what s left, and so on. Such a decomposition is shown for n = 7, d = in Figure.1. For d HJ(n ), we will have terminal drawing positions for all of the subhypercubes of dimension 1 in the decomposition. Note that, if after coloring the central [n ] d hypercube with such a terminal drawing position, we find we have a surplus in some color, we can flip the roles of the two colors in the terminal drawing positions we assign to smaller subhypercubes in the decomposition, in the hopes of balancing out the difference. At first glance, it might seem silly to expect that we could achieve a balanced coloring in this way, but Figure.1 is misleading! The important idea is that when the dimension is large, volumes of solids are concentrated on their boundaries: for example, it is easy to see that the fraction of the volume of unit n-sphere within distance ε of its boundary is 1 (1 ε) n, which tends to 1 exponentially fast. For our [n] d hypercubes, this phenomenon will mean that almost all the volume lies on the smaller faces, allowing us to perfectly balance out the coloring as we go..1 Which Halving Hales-Jewett number? In the Introduction, we defined HJ(n) as the minimum D for which d D implies that any -coloring of [n] d contains a monochromatic winning set. It is easy to see that we could have equivalently defined HJ(n) simply as the minimum D such that any -coloring of [n] D contains a monochromatic winning set. This just because when all

21 15 -colorings of [n] D contain monochromatic winning sets, it follows that all -colorings of [n] d contain monochromatic winning sets for all d D, since the [n] D hypercube is a slice of the larger [n] d hypercube. This simple equivalence no longer holds for the balanced-coloring analog of the Halving Hales-Jewett number: since a balanced coloring of [n] d is not generally balanced on a slice [n] D (D d) the presence of terminal drawing positions in the [n] D game does not seem to imply the presence of terminal drawing positions in the [n] d game. There is thus the possibility of a fuzzy threshold associated with the Halving Hales- Jewett number, where, for some fixed n, we may have, say, that there are no terminal Tic-Tac-Toe drawing positions for d < 100 or d = 103, 105, 110 while all other values of d admit such positions. This gives rise to two different notions of the Halving Hales- Jewett number: we define HJ 1 (n), as before, as the smallest dimension D such that d D implies that any -coloring of [n] d contains a monochromatic winning set, and define HJ 1 (n) simply as the smallest dimension D for which any -coloring of [n] D contains a monochromatic winning set. (In the contrived example just given, we would have HJ 1 (n) = 111, while HJ 1 (n) = 101.) Note that HJ 1 (n) HJ 1 (n). Although it might be tempting to suspect that this kind of fuzzy threshold never actually occurs, and we have HJ 1 (n) = HJ 1 such relationships may be completely beyond reach. (n) for all n (or at least that HJ 1 (n) HJ 1 (n)) proving. Proving HJ1 (n) HJ(n ) Our argument is captured in the following Lemma. Lemma.. If d ln (n 1) and there is a -coloring of the [n ]d hypercube without any monochromatic winning sets, then there is a balanced -coloring of the [n] d hypercube without any monochromatic winning sets. In a slight abuse of notation we identify [n ] as the set {, 3,..., n 1}. We proceed, as suggested earlier, by dividing the [n] d hypercube into subhypercubes [n ] f, 1 f d as follows: for each formal vector v = (v (1), v (),..., v (d) ) {1, c, n} d (c stands for center ), we define the subhypercube H v [n] d as the set of all points

22 16 (a (1),..., a (d) ) such that, for all 1 j d, a (j) = 1 if and only if v (j) = 1, and, similarly, a (j) = n if and only if v (j) = n. Referring to Figure.1, the large central [5] subhypercube in that decomposition corresponds to the formal vector (c, c), while the four [5] 1 subhypercubes (shown in gray in the Figure) correspond to the vectors (1, c), (7, c), (c, 1), and (c, 7), and the 4 corners correspond to the vectors (1, 1), (1, 7), (7, 1), (7, 7). The dimension of such subhypercubes is, of course, just the number of c s in their corresponding formal vectors, and we call subhypercubes (and their corresponding formal vectors) degenerate when their dimension is 0. Notice that the decomposition of [n] d into the subhypercubes H v mimics the terms of the binomial expansion n d = ((n )+) d = (n ) d +d (n ) d 1 + Ç å d (n ) d + +d d 1 (n )+ d. (.3) That is: the number of (f)-dimensional hypercubes in the decomposition is d f d f. The most important property of the decomposition we have described is the following: Observation.3. For any winning set W in the [n] d hypercube, there is some nondegenerate v so that W H v is a winning set in the H v hypercube. In particular, if each of the nondegenerate H v s are colored without any monochromatic winning set, then [n] d is colored without any monochromatic winning set. Proof. This is more or less intuitively obvious. Recall that a winning set W in [n] d is a sequences (c (1) j, c () j,..., c (d) j ) (1 j n) such that for any fixed k, the coordinate sequence {c (k) j } n j=1 is either constant, the increasing sequence 1,,..., n, or the decreasing sequence n, n 1,..., 1. We associate to such a sequence the formal vector v W which has 1 in the dth place whenever c (d) is the constant 1 sequence, n in the dth place whenever c (d) is the constant n sequence, and otherwise has c in the dth place. (Note that since W must have at least one moving coordinate, v W is nondegenerate.) Now it is easy to see that W H vw is a winning set in H vw. We are now ready to prove the Lemma. By our assumption, there is a -coloring of the central [n ] d hypercube which is proper (there are no monochromatic winning sets). Let α 0 be the proportion of x s in the coloring; we may assume that α 0 1 by

23 17 flipping the roles of x and o if necessary. Viewing the central [n ] d hypercube as n copies of a [n ] d 1 hypercube, there must be at least an α 0 fraction of x s in at least one of these (n ) slices ; therefore, we have that there is a proper -coloring of an [n ] d 1 hypercube with in which the fraction of x s is α 1 α 0. Continuing this argument, we see that for each 0 j d, there is a proper -coloring χ j of any [n ] d j hypercube in which the proportion of x s is α j, where α d α d 1 α 1 α 0. (.4) We will now build a balanced proper coloring of the [n] d hypercube from the (notnecessarily balanced) proper colorings χ j of the [n ] d j hypercubes. Begin by coloring the central [n ] d hypercube with χ 0. After doing this, the discrepancy of our partial coloring (the absolute value of the difference between the number of x s and the number of o s) is (α 1 1)(n ) d. We continue by coloring the (d 1)-dimensional H v s with the colorings χ 1, switching the roles of x s and o s as necessary to minimize the resulting discrepancy in the resulting partial coloring. We continue in this fashion, coloring the subhypercubes H v of successively smaller dimension with colorings, attempting to minimize the total discrepancy of the resulting coloring by switching the roles of colors when appropriate. Observation.3 guarantees the resulting coloring will be proper. When can we be guaranteed that the resulting coloring will have discrepancy 1? Since line (.4) insures that the colorings χ j are increasingly unbalanced, we can be guaranteed a balanced coloring so long as, after coloring any subhypercube H v, the number of total squares remaining to be colored is at least 1 less than the number H v of squares in H v. Dropping the 1 less allowance, we see from referring to to line (.3) that it is enough to require that, for all J, we have (n ) d J d j=j+1 Ç å d j (n ) d j. (.5) j Claim: Line (.5) is satisfied so long as d ln (n 1). Proving the claim falls naturally into two cases. In the case J = 0, the right-hand side

24 18 of (.5) is equal to n d (n ) d (by the binomial theorem) and so we are requiring n d (n ) d, which is equivalent to Å 1 + ã d, n which holds for d ln (n 1), from the inequality Ä ä m+ 1 m e for positive integers m (as can easily be proved using Calculus). For the second case J 1, we even have a stronger version of (.5): Ç å d (n ) d J J+1 (n ) d J 1 ; (.6) J + 1 that is, the leading term of the sum is sufficent to dominate (n ) d J. For 1 J < d this is because the righthand side is at least d (n ) d J 1 (n ) d j, while for J = d this follows since d (n ) for all d ln (n 1) (for small n this can be checked by hand). This completes the proof of the Claim, and of the Lemma as well. Hales and Jewett s original paper contained the simple lower bound HJ(n) n. In particular, this implies that if we set d = HJ(n ) 1, d satisfies d ln (n 1) for n 5, and so the fact that there is a proper coloring of the [n ] d hypercube implies that there is a balanced proper coloring of the [n] d hypercube, giving the desired inequality HJ 1 (n) HJ(n ) for n 5. The smaller cases follow from trivial examination; it is very easy to give a balanced coloring of a 4 board without monochromatic winning sets, for example (in fact, Player has a strong draw on this board). What about the stronger inequality HJ 1 (n) HJ(n )? The proof we have given

25 19 doesn t show this, as the possibility remains that we lack balanced proper colorings for dimensions d below the linear threshold required for the application of Lemma.. However, it is possible to deduce this inequality from the Lemma for sufficiently large n, with help from Beck s game-theoretic results on the strong draw in Tic-Tac-Toe to fill in the gaps below the bound on d required for the Lemma. This is simply because for large n, Becks lower bound d Å log 16 o(1) ã n log n on the maximum dimension in which Player has a strong draw is larger than the threshold d ln (n 1) required for the application of Lemma.. We state the result in the following theorem: Theorem.4. For sufficiently large n, we have HJ(n ) HJ 1 (n) HJ 1 (n) HJ(n). (.7).3 Further questions The most immediate consequence of Theorem.1 is that it allows us to translate the exponential lower bound (.1) into a lower bound of (n )/4 on HJ 3(n ) 4 1 (n) (and as well on HJ 1 (n) for sufficiently large n). This ensures that there are infinitely many [n] d Tic-Tac-Toe games where Player doesn t have a strong draw, yet where terminal drawing positions exist. We think of these as the interesting Tic-Tac-Toe games, since for these games, either Player can draw in spite of not being able to prevent a weak win ( delicate draw games ) or Player 1 can avoid the terminal drawing positions and win ( delicate win games ). It is embarrasing that, even though we can now assert that the union of these to classes is infinite, we cannot even prove that the classes separately have size. Another natural line of inquiry concerns the relationship between HJ 1 (n) and HJ(n). It may be extremely difficult to prove that they are equal for all n, and, the small cases notwithstanding, it is perhaps conceivable that it might not be true. Note that the

26 0 Tic-Tac-Toe hypergraph is quite assymmetric. For odd n, for example, the degree of the central square (n/, n/,..., n/) is at least three times greater than the degree of any other square for d 5. Apart from suggesting the central square as a good move in a Tic-Tac-Toe game, this suggests perhaps that there could exist [n] d hypercubes for odd n which can be properly -colored, but only when the coloring is unbalanced in favor of the color not used on the central square. Unfortunately, any of these questions might be completely intractable. The Ramsey number R(5) is often cited as a quantity which seems deceptively simple, about which we are embarrasingly ignorant, but the Hales-Jewett number is much worse! That HJ(3) = 3 can be checked by a simple case study, but for HJ(4) we don t even have an upper bound like 10 10,000. The bound for HJ(4) coming from Shelah s recursive upper bound [47] on HJ(n) is HJ(4) where the tower has height 4. On the other side of things, the best known lower bound is HJ(4) 5, due to Golomb and Hales [7].

27 1 Chapter 3 Odd-girth in dense k-critical graphs A significant result in the study of dense critical graphs was the construction by Toft of a 4-critical graph with > 1 16 n edges (Figure 3.1). To this day, this is the densest known such graph. One curious feature of Toft s graph is that it is triangle-free. Thus, when combined with Mycielski s operation (Figure 3.), this old construciton of Toft gives k-critical triangle-free graphs with 3k 4 4 k n edges. The work of this chapter is motivated by the fact that the density constants for such graphs tends to 0 (quite quickly!) as k grows large. Can they be improved upon? For the particular case of k = 5, Gyárfás gave (personal communication) the construction shown in Figure 3.3 of a 5-critical graph with n edges, showing that, at the very least, the constants we get from applications of Mycielski s operation are not best possible, since > Theorem 1.1 (restated below) shows that it is possible not just to improve these constants for all k, but to get quadratic density for k-critical triangle-fee graphs which is independent of k. Theorem 1.1 (restated). For k 4, there are triangle-free k-critical graphs with > (c k o(1))n edges, where here c , c , and c k = 1 4 for all k 6. Figure 3.1: Toft s graph is a dense, triangle-free, 4-critical graph. It is constructed by joining sets of size l + 1 in a complete bipartite graph, and matching each to a (l + 1)-cycle. It has 1 16 n + n edges.

28 x v v Figure 3.: (Mycielski s operation µ(g) on G is: create a new vertex v for each v G, join it to the neighbors of v in G, and join a new vertex x to all of the new vertices v. Shown is µ(c 5 ). This operation preserves triangle-freeness and criticality, while increasing the chromatic number by 1. (In the case k = 4 our construction is identical to Toft s graph.) Theorem 1.1 follows from our constructions in Section 3.1, which can be seen as generalizing the Toft graph. The basic idea of the construction is the following: the Toft and Gyárfás constructions work by pasting graphs onto independent sets so that in any (k 1)-coloring of the graph, at least two different colors occur in each of the independent sets (and the pasting has suitable criticality-like properties as well). In Section 3.1, we recursively define a way of critically pasting graphs to independent sets (in a triangle-free way) so that we can be assured that in a (k 1)-coloring of a graph, ö k colors occur in each independent set (i.e., ö k ù in one and k in the other). Moreover, the new pasted parts will be of negligable size compared with the independent sets. From this point, we simply defer back to the idea of Toft s construction, joining two such independent sets in a complete bipartite graph. The biggest surprise regarding Theorem 1.1 is that the constants it gives for k 6 are best possible, since triangle-free graphs have 1 4 n edges. This and Theorem Figure 3.3: Gyárfás s graph: a dense, triangle-free, 5-critical graph. Each of the three double crosses stands for the edges of the complete bipartite graph joining the independent sets on either side. (For this particular n = 161, each double cross represents 400 edges). Gyárfás s graph is dense, with ( o(1))n edges.

29 3 1., restated below, appear to be the only results known which determine the exact asymptotic density of k-critical members (k 6) of natural classes of graphs. Theorem 1. (restated). For k 4, there are pentagon-and-triangle-free k-critical graphs with > (c 5,k o(1))n edges, where c 5,4 1 36, c 5,5 3 35, and c 5,k = 1 4 for k 6. Unlike the proof of Theorem of 1.1, the proof of Theorem 1. involves a deletion argument and is thus not constructive. The proof is in Section 3., and combines the idea of our construction for Theorem 1.1 with a modified Mycielski operation. (Note that even just pentagon-free graphs have at most 1 4 n edges as a consequence of the classical Erdős-Stone theorem [5], since odd-cycles are 3-chromatic.) In light of Theorems 1.1 and 1., it is natural to consider graphs avoiding more odd cycles. (Recall that graphs avoiding any fixed even cycle have subquadratic edge density.) In general, we define each of the density constants c l,k as the supremum of constants c such that there are families of (infinitely many) k-critical graphs of oddgirth > l and with > cn edges, and, as in Theorem 1.1, abbreviate c 3,k as c k. (The odd-girth of a graph is the length of its shortest odd-cycle.) Unfortunately, our only results when l > 5 concern the case k = 4: Theorem 3.1. For each l 3, there is a dense family of (infinitely many) 4-critical graphs of odd-girth > l. In particular, c l,4 1 (l + 1). (3.1) Theorem 3.1 follows from our construction in Section 3.3 (which uses an extension of the Mycielski operation). In terms of the constants c l,k, the results of Theorems 1.1, 1., and 3.1 are summarized in Table 3.1 (in Section 3.4). 3.1 Avoiding triangles (constructions) To construct critical triangle-free graphs, we will construct graphs U which have large independent sets (essentially as large as U for k 6 in fact) in which ö k colors must show up in a (k 1)-coloring. (In fact, our techniques would allow us to require that

30 4 S 1 S Figure 3.4: Constructing U(S 1, S ) from the graphs S 1 and S. The rectangle encloses the set of active vertices. (Edges within the S i are not drawn.) all k 1 colors show up in the independent set, but this is not the best choice from a density perspective.) The U s will have criticality-like properties as well (Lemma 3.6), and we will be able to join the relevant independent sets from two such U s in a complete bipartite graph to get our construction. Let us now specify the construction of the sets U. Given graphs S 1, S,... S t, we construct U(S 1, S,..., S t ) as follows: take the (disjoint) union of the graphs S i, together with the independent set A = V (S i ), which we call the active set of vertices, and join vertices v S i to vertices u A which equal v in their i th coordinate. This construction is illustrated for t = in Figure 3.4. Where r 1, r,... r t are positive integers, we write U(r 1, r,... r t ) to mean a construction U(S 1, S,..., S t ) where, for each i, S i is a triangle-free r i -critical graph without isolated vertices. With just one graph S, U(S) simply consists of a matching between S and an independent set of the same size. Thus the Toft graph is two identical copies of a U(3), whose active sets are joined in a complete bipartite graph. Gyárfás s construction is now four copies C 1, C, C 3, C 4 of a U(4), where the active sets of C i and C i+1 are joined in a complete bipartite graph for 1 i 3, and the active sets of both C 1 and C 4 are completely joined to a new vertex x. Observation 3.. U(r 1, r,..., r t ) is triangle-free, and if s i = S i, it has s 1 s t active vertices, and s i structural vertices of type i.

31 5 The structural vertices of type i are vertices from the copies of S i used in the construction of U(r 1, r,... r t ). Our inductive proof of Lemma 3.6 will use the following properties of the sets U(S 1,..., S t ): Observation 3.3. Let A ω S i be the all the vertices of the active set whose tth coordinate is the vertex ω S t. Then the subgraph of U(S 1,... S t ) induced by the set A ω V (S i ) is isomorphic to U(S 1,..., S t 1 ). 1 i<t Observation 3.4. Any colorings of the graphs S 1, S,... S t can be extended to the rest of U(S 1, S,..., S t ) ( i.e., to the active vertices) so that just t + 1 different colors of our choosing appear in the active set. Proof. This follows from observing that the degree of each active vertex in U(S 1,..., S t ) is t. Before proceeding any further, we point out the following basic fact, which lies at the heart of all recursive constructions of critical graphs: Observation 3.5. A k-critical graph without isolated vertices can be k-colored so that color k occurs only at a single vertex of our choice. Proof. This follows from observing that such a graph is vertex-critical; i.e., removing any vertex decreases the chromatic number. We are now ready to prove the main lemma, about the sets Uk i+1 k 1 U(k 1, k,..., k i + 1). Basically, it says that in a (k 1)-coloring, their active sets have a set of properties similar to criticality. Lemma 3.6. The sets Uk i+1 k 1 = U(k 1, k,..., k i + 1) satisfy: 1. Uk i+1 k 1 can be properly (k 1)-colored so that i different colors appear at active vertices and so that among active vertices, only one vertex of our choosing gets the ith color.

32 6. In any (k 1)-coloring of Uk i+1 k 1, at least i different colors occur as the colors of active vertices. 3. If any edge from U k 1 k i+1 most i 1 colors occur at active vertices. is removed, it can be properly (k 1)-colored so that at Proof. We prove Lemma 3.6 by induction on i. For i = 1, we interpret U k 1 k as simply a single active vertex and the statement is trivial. Recall that the graphs S j (1 j i 1) are the triangle-free (k j)-critical graphs used in the construction of Uk i+1 k 1 k 1. For i > 1, Observation 3.3 tells us that Uk i+1 consists of s i 1 = S i 1 copies C ω of a U k 1 k i+, one for each vertex ω S i 1, which are pairwise disjoint except at the graphs S 1, S,..., S i (in other words, their active sets A(C ω ) are disjoint). Note that each ω S i 1 is adjacent to every vertex in A(Cω). To prove part 1, color S i 1 with colors i 1 through k 1 so that color i 1 occurs only at vertex ω 0 (we can do this by Observation 3.5). By induction, (k 1)-color C ω0 so that its active vertices get colors from 1,,..., i, i and so that color i occurs at only one vertex u of C ω0. Now we can use Observation 3.4 to extend the current partial (k 1)-coloring to the rest of the C ω s so that their active vertices get colors from 1 through i 1; this gives us a (k 1)-coloring of Uk i+1 k 1 where active vertices get colors from 1 through i, and i occurs only at u. Finally, note that we could have chosen so that u was any active vertex. For part, notice that by induction, i 1 different colors occur at active vertices of the C ω s in a (k 1)-coloring of Uk i+1 k 1. Thus if only i 1 colors appear at active vertices of Uk i+1 k 1, the same set of i 1 colors occurs at the sets of active vertices of each of the C ω s; but then S i is colored with a disjoint set of colors, but then we need (i 1) + (k i + 1) > k 1 colors overall, a contradiction. For part 3, assume first that the removed edge came from S i : Then we can (k i)- color what remains, and (inductively by part 1) color the C ω s so that only the leftover i 1 colors occur at the active vertices. If the removed edge was one joining a vertex ω 0 in S i to a vertex v in the active set of C ω0, we color S i with the colors i 1 through k 1 so that color i 1 occurs only at