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For two graphs B and H the strong Ramsey game R(B,H) on the board B and with target H is played as follows. Two players alternately claim edges of B. The first player to build a copy of H wins. If none of the players win, the game is declared a draw. A notorious open question of Beck (1996, 2002, 2011) asks whether the first player has a winning strategy in R(Kn,Kk) in bounded time as n→∞. Surprisingly, in a recent paper Hefetz et al. (2017) constructed a 5-uniform hypergraph H for which they proved that the first player does not have a winning strategy in R(Kn(5),H) in bounded time. They naturally ask whether the same result holds for graphs. In this paper we make further progress in decreasing the rank. In our first result, we construct a graph G (in fact G=K6∖K4) and prove that the first player does not have a winning strategy in R(Kn⊔Kn,G) in bounded time. As an application of this result we deduce our second result in which we construct a 4-uniform hypergraph G′ and prove that the first player does not have a winning strategy in R(Kn(4),G′) in bounded time. This improves the result in the paper above. By compactness, an equivalent formulation of our first result is that the game R(Kω⊔Kω,G) is a draw. Another reason for interest on the board Kω⊔Kω is a folklore result that the disjoint union of two finite positional games both of which are first player wins is also a first player win. An amusing corollary of our first result is that at least one of the following two natural statements is false: (1) for every graph H, R(Kω,H) is a first player win; (2) for every graph H if R(Kω,H) is a first player win, then R(Kω⊔Kω,H) is also a first player win. Surprisingly, we cannot decide between the two.

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... Specifically, they proved the existence of a 5-uniform hypergraph H such that, in the game RpK p5q n , Hq, P 1 cannot secure a win within a bounded number of moves. Subsequently, in [8], David, Hartarsky, and Tiba investigated another intriguing generalization where the board consists of two vertex-disjoint copies of K n . They demonstrated that when H " K 6 zK 4 , P 1 cannot win the game RpK n \ K n , Hq within a bounded number of moves. ...
... • For graph (6), since (3) and (6) are isomorphic, P 1 can also win within 6 rounds. (7) (8) and (9) Case 3. Proofs of (7), (8) and (9). ...
... Proof. For graphs (7), (8) and (9), P 2 can claim two more edges. Clearly, no matter how P 2 claims those two edges, P 2 cannot obtain a K 2,2 p1q, which implies that P 2 is not in check. ...
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The strong Ramsey game R(B,H)R(\mathcal{B}, H) is a two-player game played on a graph B\mathcal{B}, referred to as the board, with a target graph H. In this game, two players, P1P_1 and P2P_2, alternately claim unclaimed edges of B\mathcal{B}, starting with P1P_1. The goal is to claim a subgraph isomorphic to H, with the first player achieving this declared the winner. A fundamental open question, persisting for over three decades, asks whether there exists a graph H such that in the game R(Kn,H)R(K_n, H), P1P_1 does not have a winning strategy in a bounded number of moves as nn \to \infty. In this paper, we shift the focus to the variant R(KnKn,H)R(K_n \sqcup K_n, H), introduced by David, Hartarsky, and Tiba, where the board KnKnK_n \sqcup K_n consists of two disjoint copies of KnK_n. We prove that there exist infinitely many graphs H such that P1P_1 cannot win in R(KnKn,H)R(K_n \sqcup K_n, H) within a bounded number of moves through a concise proof. This perhaps provides evidence for the existence of examples to the above longstanding open problem.
... In Chapter 10, we make further progress on a question of Beck [19] and a question of Hefetz, Kusch, Narins, Pokrovskiy, Requilé and Sarid [94] on the existence of Ramsey Games in which the first player has a winning strategy but not in bounded time. The work in this chapter is joint with Stefan David and Ivailo Hartarsky and is published in the European Journal of Combinatorics [50]. ...
... The work in this section is joint with Stefan David and Ivailo Hartarsky and is published in the European Journal of Combinatorics [50]. ...
Thesis
This thesis consists of an introduction and nine chapters, each devoted to a different combinatorial problem. What gives these problems coherence is that although at first sight they look very different, the essential difficulties in most are geometric. In Chapters 2 and 3, we investigate sumset inequalities. For sets A,B in an abelian group G, define the sumset A+B={a+b : aA,bB}A+B=\{a+b\text{ : }a \in A, b\in B\}. We consider the ambient groups Zk\mathbb{Z}^k and Rk\mathbb{R}^k, endowed with the measure |\cdot| representing cardinality and outer Lebesgue measure, respectively. One of the central questions in additive combinatorics is the inverse sumset problem of characterizing the finite subsets A with small doubling constant\textit{doubling constant} A+AA1|A+A|\cdot|A|^{-1}. Sets A in Rk\mathbb{R}^k have doubling constant at least 2k2^k; this is no longer true for sets A in Zk\mathbb{Z}^k, unless some non-degeneracy\textit{non-degeneracy} condition is imposed. Our main result describes the structure of non-degenerate\textit{non-degenerate} sets A in Zk\mathbb{Z}^k with doubling constant close to 2k2^k. We prove a sharp stability result for the classical Freiman--Bilu 2k2^k-inequality (improved by Green--Tao) and a generalization of Freiman's 3k43k-4 theorem to arbitrary dimension. For δ>0\delta>0 sufficiently small and AZkA\subset \mathbb{Z}^k with A+A(2k+δ)A|A+A|\le (2^k+\delta)|A|, we show either A is covered by mk(δ)m_k(\delta) parallel hyperplanes, or it satisfies |\coo(A)\setminus A|\le c_k\delta |A|, where \coo(A) is the smallest convex progression (convex set intersected with an affine sub-lattice) containing A. In particular, we deduce the sharp stability result for the Brunn--Minkowski inequality for equal sets, conjectured by Figalli and Jerison. Given δ>0\delta>0 sufficiently small and ARkA\subset \mathbb{R}^k with A+A(2k+δ)A|A+A|\le (2^k+\delta)|A|, we show |\co(A)\setminus A|\le c_k\delta |A|, where \co(A) is the smallest convex set containing A. Finally, we prove a strengthen version of the aforementioned conjecture by Figalli and Jerison for a specific class of geometric objects. We find the optimal constants ckc_k, when ARkA \subset \mathbb{R}^k is the hypograph of a function defined on a convex domain in dimension k13k-1 \leq 3. In Chapters 4, 5 and 6, we investigate covering systems. A covering system is a finite collection of arithmetic progressions \{a_1\text{ }(\text{mod } m_1),a_2\text{ }(\text{mod } m_2), \hdots, a_k\text{ }(\text{mod } m_k) \} that cover the integers, i.e., i{ai+nmi : nZ}=Z\cup_i \{a_i+ n m_i \text{ : } n \in \mathbb{Z}\}=\mathbb{Z}. Since their introduction by Erdős in 1950, covering systems have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. More than fifty years ago, Erdős asked if the moduli can be distinct and all arbitrarily large, Erdős and Selfridge asked if the moduli can be distinct and all odd, and Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. Another beautiful conjecture, proposed by Erdős and Graham in 1980, states that if the moduli are distinct elements of the interval [n,Cn], and n is sufficiently large, then the density of integers uncovered by the union is bounded below by a constant (depending only on~C). This conjecture was confirmed (in a strong form) by Filaseta, Ford, Konyagin, Pomerance and Yu in 2007, who moreover asked whether the same conclusion holds if the moduli are distinct and sufficiently large, and i=1k1di<C\sum_{i=1}^k \frac{1}{d_i} < C. Although, as it turns out, this condition is not sufficiently strong to imply the desired conclusion, as one of the main results of this paper we give an essentially best possible condition which is sufficient. More precisely, we show that if all of the moduli are sufficiently large, then the union misses a set of density at least e4C/2e^{-4C}/2, where C=i=1kμ(di)di C = \sum_{i=1}^k \frac{\mu(d_i)}{d_i} and μ\mu is a multiplicative function defined by \mu(p^i)=1+(\log p)^{3+\eps}/p for some \eps > 0. We also show that no such lower bound (i.e., depending only on~C) on the density of the uncovered set holds, when μ(pi)\mu(p^i) is replaced by any function of the form 1+O(1/p). Our method has a number of further applications. Most importantly, we prove the conjecture of Schinzel. In addition, we give an alternative (somewhat simpler) proof of a breakthrough result of Hough, who resolved Erdős' minimum modulus problem, with an improved bound on the smallest difference. Moreover, we make further progress on the problem of Erdős and Selfridge, which, in particular, we solve in the square free case. Finally, we answer another natural question of Erdős, asked in 1952, on the \emph{number} of minimal covering systems. Addressing this question requires very different methods. %, that is, covering systems such that the removal of any progression leaves an element uncovered. More precisely, we show that the number of minimal covering systems with exactly n elements is exp((4τ3+o(1))n3/2(logn)1/2) \exp\left( \left(\frac{4\sqrt{\tau}}{3} + o(1)\right) \frac{n^{3/2}}{(\log n)^{1/2}} \right) as nn \to \infty, where τ=t=1(logt+1t)2. \tau = \sum_{t = 1}^\infty \left( \log \frac{t+1}{t} \right)^2. \textit{En route} to this counting result, we obtain a structural description of all covering systems that are close to optimal in an appropriate sense. Chapter 7 is devoted to Littlewood polynomials. Answering a question of Erd\H{o}s from 1957 and confirming a conjecture of Littlewood from 1966, we show that there exist absolute constants Δ>δ>0\Delta > \delta > 0 such that, for all n2n \ge 2, there exists a polynomial P of degree~n, with coefficients in {1,1}\{-1,1\}, such that δnP(z)Δn \delta\sqrt{n} \le |P(z)| \le \Delta\sqrt{n} for all z\in\C with z=1|z|=1. Over time, this problem attracted considerable attention and, in particular, Littlewood included it in his well-known monograph containing 30 of his favourite problems. Chapter 8 is devoted to judicious partitions. Bollobás, Reed and Thomason proved that every 3-uniform hypergraph with m hyperedges has a vertex-partition into 3 parts such that each part meets at least 13(11e)m\frac{1}{3}(1-\frac{1}{e})m hyperedges. Halsegrave optimized the value to 0.6m, confirming a special case of a conjecture of Bollobás and Thomason from 1993. For large values of m, Ma and Yu improved the bound asymptotically to 0.65m+o(m). We further improve this asymptotic bound to 1927m+o(m)\frac{19}{27}m+o(m), which is best possible up to the error term, resolving the next open case of a conjecture of Bollobás and Scott from 2000. Chapter 9 is devoted to coloured structures in the Boolean lattice. Given a collection of coloured chain posets, we estimate the number of coloured subsets of the Boolean lattice which avoid all chains in this collection. Our proof relies on the recent powerful method of hypergraph containers developed independently by Balogh, Morris and Samotij as well as Saxton and Thomason, inspired by the previous work of Conlon and Gowers. In order to prove results about coloured chain posets we need to apply the hypergraph container lemma recursively, with different uniformities at each stage, using a balanced supersaturation result for a certain non-uniform hypergraph encoding forbidden configurations. Our work extends to a coloured setting the previous work of Collares and Morris as well as Balogh, Mycroft, and Treglown on the number of antichains in a random subset of the Boolean lattice. Chapter 10 is devoted to positional games. For two graphs B and H the strong Ramsey game R(B,H)\mathcal{R}(B,H) on the board B and with target H is played as follows. Two players alternately claim edges of B. The first player to build a copy of H wins. If none of the players win, the game is declared a draw. A notorious open question of Beck asks whether the first player has a winning strategy in R(Kn,Kk)\mathcal{R}(K_n,K_k) in bounded time as nn\rightarrow\infty. Surprisingly, in a recent paper Hefetz, Kusch, Narins, Pokrovskiy, Requilé and Sarid constructed a 5-uniform hypergraph H\mathcal{H} for which they proved that the first player does not have a winning strategy in R(Kn(5),H)\mathcal{R}(K_n^{(5)},\mathcal{H}) in bounded time. They naturally asked whether an analogous result holds for graphs. We make further progress towards this question.
... They are included, since they are tightly linked to the remainder of the study. Instead, we do not include works unrelated to our main subject, completed both prior to the beginning of the thesis [115,206,211] and during its course [212]. ...
... Again, by Lemma 4.2.7 it is not hard to see that if B and T i I for i I i are fully infected and GpT i q occurs, then the U-bootstrap percolation can also infect T i (and similarly for T ¡ i ). 115 Finally, the super good event SGpV q is dened as SGpBq i pGpT i q GpT ¡ i qq and it clearly implies that the entire snail V can be infected from within. ...
Thesis
We study two tightly related classes of statistical mechanics models on the two-dimensional square lattice—kinetically constrained models and bootstrap percolation. The former arose as models of the dynamics of supercooled liquids close to the glass transition, while the latter are used to model a number of settings including magnets and social phenomena. We consider both kinetically constrained models and bootstrap percolation from a rigorous probabilistic perspective. We are interested in their behaviour as their parameter approaches its (possibly degenerate) critical value. More specifically, we investigate the rate of divergence of certain characteristic time scales, such as the infection time of a fixed site and the relaxation time. Among the highlights of the thesis is determining the universality classes of kinetically constrained models together with their characteristic equilibrium time scales at low temperature. That is, we establish a partition of all possible models into groups with similar behaviour and provide a recipe for determining the behaviour from the definition of the model. Contributions are made to the full spectrum of universality classes of kinetically constrained models, but in some cases also to the simpler and better understood bootstrap percolation. In addition to universal results, we provide sharp asymptotics in both bootstrap percolation and kinetically constrained models for the most classical, 2-neighbour, model, as well as advances on the 1-neighbour kinetically constrained model known as the Fredrickson–Andersen 1-spin facilitated model. The thesis consists of three main parts based on techniques from different domains. The first one relates to dynamics of interacting particle systems. The second one relies on combinatorial arguments. The third and final part takes a percolation viewpoint.
... David et al. [3] noted that according to a comprehensive strategy-stealing argument by Nash, Bob cannot possess a winning strategy in the strong Ramsey games. Applying Ramsey's theorem [10] to any fixed target graph G, it is established that for sufficiently large n, the Ramsey number R (K n , G) does not result in a draw. ...
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Let p,q be two integers with pqp\geq q. Given a finite graph F with no isolated vertices, the generalized Ramsey achievement game of F on the complete graph KnK_n, denoted by (p,q;Kn,F,+)(p,q;K_n,F,+), is played by two players called Alice and Bob. In each round, Alice firstly chooses p uncolored edges e1,e2,...,epe_1,e_2,...,e_p and colors it blue, then Bob chooses q uncolored edge f1,f2,...,fqf_1,f_2,...,f_q and colors it red; the player who can first complete the formation of F in his (or her) color is the winner. The generalized achievement number of F, denoted by a(p,q;F){a}(p,q;F) is defined to be the smallest n for which Alice has a winning strategy. If p=q=1, then it is denoted by a(F){a}(F), which is the classical achievement number of F introduced by Harary in 1982. If Alice aims to form a blue F, and the goal of Bob is to try to stop him, this kind of game is called the first player game by Bollob\'{a}s. Let a(F){a}^*(F) be the smallest positive integer n for which Alice has a winning strategy in the first player game. A conjecture due to Harary states that the minimum value of a(T){a}(T) is realized when T is a path and the maximum value of a(T){a}(T) is realized when T is a star among all trees T of order n. He also asked which graphs F satisfy a(F)=a(F)a^*(F)=a(F)? In this paper, we proved that na(p,q;T)n+q(n2)/pn\leq {a}(p,q;T)\leq n+q\left\lfloor (n-2)/p \right\rfloor for all trees T of order n, and obtained a lower bound of a(p,q;K1,n1){a}(p,q;K_{1,n-1}), where K1,n1K_{1,n-1} is a star. We proved that the minimum value of a(T){a}(T) is realized when T is a path which gives a positive solution to the first part of Harary's conjecture, and a(T)2n2{a}(T)\leq 2n-2 for all trees of order n. We also proved that for n3n\geq 3, we have 2n2(4n8)ln(4n4)a(K1,n1)2n22n-2-\sqrt{(4n-8)\ln (4n-4)}\leq a(K_{1,n-1})\leq 2n-2 with the help of a theorem of Alon, Krivelevich, Spencer and Szab\'o. We proved that a(Pn)=a(Pn)a^*(P_n)=a(P_n) for a path PnP_n.
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We consider the strong Ramsey-type game R(k)(H,0)\mathcal{R}^{(k)}(\mathcal{H}, \aleph_0), played on the edge set of the infinite complete k-uniform hypergraph KNkK^k_{\mathbb{N}}. Two players, called FP (the first player) and SP (the second player), take turns claiming edges of KNkK^k_{\mathbb{N}} with the goal of building a copy of some finite predetermined k-uniform hypergraph H\mathcal{H}. The first player to build a copy of H\mathcal{H} wins. If no player has a strategy to ensure his win in finitely many moves, then the game is declared a draw. In this paper, we construct a 5-uniform hypergraph H\mathcal{H} such that R(5)(H,0)\mathcal{R}^{(5)}(\mathcal{H}, \aleph_0) is a draw. This is in stark contrast to the corresponding finite game R(5)(H,n)\mathcal{R}^{(5)}(\mathcal{H}, n), played on the edge set of Kn5K^5_n. Indeed, using a classical game-theoretic argument known as \emph{strategy stealing} and a Ramsey-type argument, one can show that for every k-uniform hypergraph G\mathcal{G}, there exists an integer n0n_0 such that FP has a winning strategy for R(k)(G,n)\mathcal{R}^{(k)}(\mathcal{G}, n) for every nn0n \geq n_0.
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For a positive integer k we consider the k-vertex-connectivity game, played on the edge set of KnK_n, the complete graph on n vertices. We first study the Maker-Breaker version of this game and prove that, for any integer k2k \geq 2 and sufficiently large n, Maker has a strategy for winning this game within kn/2+1\lfloor k n/2 \rfloor + 1 moves, which is clearly best possible. This answers a question of Hefetz, Krivelevich, Stojakovi\'c and Szab\'o. We then consider the strong k-vertex-connectivity game. For every positive integer k and sufficiently large n, we describe an explicit first player's winning strategy for this game.
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This is a partly survey, partly new results paper about Ramsey games. Ramsey games belong to the wider class of positional games. In Section 1 we briefly recall the basic concepts and results of positional games in general, and apply them to the particular case of Ramsey games. For a more detailed introduction to the theory of positional games, including proofs, we refer the reader to Beck [4] and [5]. (C) 2002 Published by Elsevier Science B.V.
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Consider the following games played on the complete graph K p by two players called Alpha and Beta. First, Alpha colors one of the edges of K p green, then Beta colors a different edge red, and so forth. A graph F with no isolated points is given. The first player who can complete the formation of F in his color is the winner. The minimum p for which Alpha can win regardless of the moves made by Beta is the achievement number of F. In the corresponding misère game, the player who first forms F in his color is the loser and the minimum p for which Beta can force Alpha to lose is the avoidance number of F. Several variations on these games are proposed, and partial results presented.
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Let us consider the following 2-player game, calledvan der Waerden game. The players alternately pick previously unpicked integers of the interval {1, 2, ...,N}. The first player wins if he has selected all members of ann-term arithmetic progression. LetW*(n) be the least integerN so that the first player has a winning strategy. By theRamsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the completek-uniform hypergraph ofN verticesK N k , and the first player wins if he has selected allk-tuples of ann-set. LetR k*(n) be the least integerN so that the first player has a winning strategy. We prove (W* (n))1/n → 2,R 2*(n) n andR k * n nk / k! fork ≧3.
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This is the third piece of a series of papers about a new, quasiprobabilistic theory of positional games (i.e., “combinatorial games”). The strange thing is that we wrote these papers in reverse order. Chronologically the first paper, “Deterministic Graph Games and a Probabilistic Intuition,” was a highly technical one, and can be considered as part III of the series. The next paper, “Achievement Games and the Probabilistic Method,” was a survey that attempted to explain the role of the subject in discrete mathematics. We felt, however, that we did not quite succeed, and somehow the foundations were not very solid. This is why we had to write this paper, which should be considered as part I of the series. Our main object here is to explain what the basic questions of positional game theory are. The well-known algebraic theory of Nim-like games (called “combinatorial game theory”) and the quasiprobabilistic theory represent two entirely different viewpoints, and they in some sense complement each other. Indeed, combinatorial game theory (i.e., Nim-like games) is an exact local theory in the sense how seemingly complicated games start out as composites, or quickly develop into composites of several simple local games. On the other hand, the quasi-probabilistic theory attempts to solve “hopelessly complicated” Tic-Tac-Toe-like games which usually remain as single coherent entities throughout play. It is an efficient global approach which, roughly speaking, evaluates via loss probabilities. Because of the intractable complexity of the exhausting search through the game-tree, an efficient evaluation method has to approximate. So one cannot really expect from the quasi-probabilistic theory to solve evenly balanced “head-to-head games,” where a single mistake could be fatal, but it can effectively recognize and solve large classes of difficult “one-sided games.” Positional games are finite 2-player games of skill (i.e., no chance moves) with perfect information, and the payoff function has three values 1, 0, −1 only (“win,” “draw,” “loss”). These games, therefore, are deterministic, and because of the perfect information, the optimal strategies are deterministic. How can randomness then enter the story? To answer this, we very briefly summarize the simplest case of the quasi-probabilistic theory: the majority principle. The majority principle is in two parts. The first part is a probabilistic intuition that says in a nutshell that, in many complicated games, the outcome between two perfect players is the same as the “majority outcome” between two “random players” (random game). The point is that even relatively simple games are too hopelessly complicated to analyze in full depth, but to describe the “typical” behavior is usually a tractable problem to solve by using probability theory. However, the majority principle is more than merely predicting the outcomes of complicated games. The second part is to convert the probabilistic intuition, via potential techniques, into effective deterministic strategies, in fact, greedy algorithms. © 1996 John Wiley & Sons, Inc.
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We consider a "Maker-Breaker" version of the Ramsey Graph Game, RG(n), and present a winning strategy for Maker requiring at most (n 0 3)2 n01 + n + 1 moves. This is the fastest winning strategy known so far. We also demonstrate how the ideas presented can be used to develop winning strategies for some related combinatorial games. Keywords: Combinatorial Games, Algorithms on Graphs, Ramsey Theory 1 Introduction The Ramsey Graph Game, RG(n), on a complete graph on N vertices, KN , is considered. Two players, Maker (red) and Breaker (blue) alternately color the edges of KN . Maker is first to play, and the players color one edge per move. Maker wins the game if there is a red Kn . Breaker wins if there is no red Kn after all the N (N 0 1)=2 edges have been colored. Let R(n; n) denote the n-th Ramsey number, i.e., the smallest number R such that for every two-coloring of the edges of KR there exsist a monochromatic Kn ae KR . Maker has a simple winning strategy if N 2R(n; n). Note ...
Combinatorial games, volume 114 of Encyclopedia of Mathematics and its Applications Tic-tac-toe theory
  • J Beck
J. Beck. Combinatorial games, volume 114 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2011. Tic-tac-toe theory, Paperback edition of the 2008 original.
Combinatorics, graph theory and computing
  • J Beck
J. Beck. Ramsey games. Discrete Math., 249(1-3):3-30, 2002. Combinatorics, graph theory and computing (Louisville, KY, 1999).