Ramsey Games Research Articles

Online size Ramsey numbers: Path vs C4

Given two graphs G and H, a size Ramsey game is played on the edge set of KN. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of G or a blue copy of H as soon as possible. The online (size) Ramsey number r˜(G,H) is the number of rounds in the game provided Builder and Painter play optimally. We prove that r˜(C4,Pn)≤2n−2 for every n≥8. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get r˜(C4,Pn)=2n−2 for n≥8. Our proof for n≤13 is computer-assisted. The bound r˜(C4,Pn)≤2n−2 solves also the “all cycles vs. Pn” game for n≥8 – it implies that it takes Builder 2n−2 rounds to force Painter to create a blue path on n vertices or any red cycle.

Off-diagonal online size Ramsey numbers for paths

Consider the following Ramsey game played on the edge set of KN. In every round, Builder selects an edge and Painter colours it red or blue. Builder’s goal is to force Painter to create a red copy of a path Pk on k vertices or a blue copy of Pn as soon as possible. The online (size) Ramsey number r̃(Pk,Pn) is the number of rounds in the game provided Builder and Painter play optimally. We prove that r̃(Pk,Pn)≤(5/3+o(1))n provided k=o(n) and n→∞. We also show that r̃(P4,Pn)≤⌈7n/5⌉−1 for n≥10, which improves the upper bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo and implies their conjecture that r̃(P4,Pn)=⌈7n/5⌉−1.

Bounds on Ramsey games via alterations

AbstractWe present a refinement of the classical alteration method for constructing ‐free graphs for suitable edge‐probabilities , we show that removing all edges in ‐copies of the binomial random graph does not significantly change the independence number. This differs from earlier alteration approaches of Erdős and Krivelevich, who obtained similar guarantees by removing one edge from each ‐copy (instead of all of them). We demonstrate the usefulness of our refined alternation method via two applications to online graph Ramsey games, where it enables easier analysis.

Online and Connected Online Ramsey Numbers of a Matching versus a Path

The (G1,G2)-online Ramsey game is a two-player turn-based game between a builder and a painter. Starting from an empty graph with infinite vertices, the builder adds a new edge in each round, and the painter colors it red or blue. The builder aims to force either a red copy of G1 or a blue copy of G2 in as few rounds as possible, while the painter’s aim is the opposite. The online Ramsey number r˜(G1,G2) is the minimum number of edges that the builder needs to win the (G1,G2)-online Ramsey game, regardless of the painter’s strategy. Furthermore, we initiate the study of connected online Ramsey game, which is identical to the usual one, except that at any time the graph induced by all edges should be connected. In this paper, we show a general bound of the online Ramsey number of a matching versus a path and determine its exact value when the path has an order of three or four. For the connected version, we obtain all connected online Ramsey numbers of a matching versus a path.

A Note on On-Line Ramsey Numbers of Stars and Paths

An on-line Ramsey game is a game between two players, Builder and Painter, on an initially empty graph with unbounded set of vertices. In each round, Builder selects two vertices and joins them with an edge while Painter colours the edge immediately with either red or blue. Builder’s aim is to force either a red copy of a fixed graph G or a blue copy of a fixed graph H. The game ends with Builder’s victory when Builder manages to force either a red G or a blue H. The minimum number of rounds for Builder to win the game, regardless of Painter’s strategy, is the on-line Ramsey number $${\tilde{r}}(G,H)$$ . This note focuses on the case when G and H are stars and paths. In particular, we will prove the upper bound of $${\tilde{r}}(S_3,P_{l+1})\le 5l/3+2$$ . We will also present an alternative proof for the upper bound of $${\tilde{r}}(S_2 = P_3, P_{l+1}) = \lceil 5l/4\rceil $$ .

Ramsey, Paper, Scissors

We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on n vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least s. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants 0 < A < B such that (under optimal play) Proposer wins with high probability if , while Decider wins with high probability if . This is a factor of larger than the lower bound coming from the off‐diagonal Ramsey number r(3,s).

Strong Ramsey games in unbounded time

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.

An upper bound for the restricted online Ramsey number

The restricted (m,n;N)-online Ramsey game is a game played between two players, Builder and Painter. The game starts with N isolated vertices. Each turn Builder picks an edge to build and Painter chooses whether that edge is red or blue, and Builder aims to create a red Km or blue Kn in as few turns as possible. The restricted online Ramsey number r̃(m,n;N) is the minimum number of turns that Builder needs to guarantee her win in the restricted (m,n;N)-online Ramsey game. We show that if N=r(n,n), r̃(n,n;N)≤N2−Ω(NlogN),motivated by a question posed by Conlon, Fox, Grinshpun and He. The equivalent game played on infinitely many vertices is called the online Ramsey game. As almost all known Builder strategies in the online Ramsey game end up reducing to the restricted setting, we expect further progress on the restricted online Ramsey game to have applications in the general case.

On two theorems of positional games

I give fully detailed proofs of two important theorems—the exact solution of the weak clique game and the compactness theorem—in the theory of positional games. Both results were published several years ago, including an outline of the proofs that explained the basic idea, but left some technical details to the reader. Unfortunately, in both cases these details turned out to be highly non-trivial. Several mathematicians asked me for help; asked me to clarify the missing details. This is why I felt obliged to write this paper.

Online Ramsey theory for a triangle on ‐free graphs

AbstractGiven a class of graphs and a fixed graph , the online Ramsey game for H on is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in , and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of forever, and we say Painter wins the game. We initiate the study of characterizing the graphs such that for a given graph , Painter wins the online Ramsey game for on ‐free graphs. We characterize all graphs such that Painter wins the online Ramsey game for on the class of ‐free graphs, except when is one particular graph. We also show that Painter wins the online Ramsey game for on the class of ‐minor‐free graphs, extending a result by Grytczuk, Hałuszczak, and Kierstead [Electron. J. Combin. 11 (2004), p. 60].

Strong Ramsey games: Drawing on an infinite board

Consider the following strong Ramsey game. Two players take turns in claiming a previously unclaimed edge of the complete graph on n vertices until all edges have been claimed. The first player to build a copy of Kq (q≥3) is declared the winner of the game. If none of the players win, then the game ends in a draw. A simple 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 that the game cannot end in a draw and is thus a first player win. A famous question of Beck asks whether the minimum number of moves needed for the first player to win this game on Kn grows with n. This seems unlikely but is still wide open. A striking equivalent formulation of this question is whether the same game played on the infinite complete graph is a first player win or a draw.The target graph of the strong Ramsey game does not have to be Kq: it can be any predetermined fixed graph. In fact, it can even be a k-uniform hypergraph (and then the game is played on the infinite k-uniform hypergraph). Since strategy stealing and Ramsey's Theorem still apply, one can ask the same question: is this game a first player win or a draw? The same intuition which led most people (including the authors) to believe that the Kq strong Ramsey game on the infinite board is a first player win, would also lead one to believe that the H strong Ramsey game on the infinite board is a first player win for any uniform hypergraph H. However, in this paper we construct a 5-uniform hypergraph for which the corresponding game is a draw.

Online Ramsey games for more than two colors

ABSTRACTConsider the following one‐player game played on an initially empty graph with n vertices. At each stage a randomly selected new edge is added and the player must immediately color the edge with one of r available colors. Her objective is to color as many edges as possible without creating a monochromatic copy of a fixed graph F. We use container and sparse regularity techniques to prove a tight upper bound on the typical duration of this game with an arbitrary, but fixed, number of colors for a family of 2‐balanced graphs. The bound confirms a conjecture of Marciniszyn, Spöhel and Steger and yields the first tight result for online graph avoidance games with more than two colors. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 464–492, 2017

On the Computational Complexity and Strategies of Online Ramsey Theory

An online Ramsey game (G,H) is a game between Builder and Painter, alternating in turns. In each round Builder draws an edge and Painter colors it either red or blue. Builder wins if after some round there is a monochromatic copy of the graph H, otherwise Painter is the winner. The rule for Builder is that after each his move the resulting graph must belong to the class of graphs G. In this abstract we investigate the computational complexity of the related decision problem and we show that it is PSPACE-complete. Moreover, we study a generalization of online Ramsey game for hypergraphs and we provide a result showing that Builder wins the online Ramsey game for the case when G and H are 3-uniform hyperforests and H is 1-degenerate.

A Remark on the Tournament Game

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker, claim unclaimed edges of $G$ in turns, while Maker additionally assigns orientations to the edges that she claims. If by the end of the game Maker claims all the edges of a pre-defined goal tournament, she wins the game. Given a tournament $T_k$ on $k$ vertices, we determine the threshold bias for the $(1:b)$ $T_k$-tournament game on $K_n$. We also look at the $(1:1)$ $T_k$-tournament game played on the edge set of a random graph ${\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker's win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied.

Online Ramsey Theory for Planar Graphs

An online Ramsey game $(G,\mathcal{H})$ is a game between Builder and Painter, alternating in turns. During each turn, Builder draws an edge, and Painter colors it blue or red. Builder's goal is to force Painter to create a monochromatic copy of $G$, while Painter's goal is to prevent this. The only limitation for Builder is that after each of his moves, the resulting graph has to belong to the class of graphs $\mathcal{H}$. It was conjectured by Grytczuk, Hałuszczak, and Kierstead (2004) that if $\mathcal{H}$ is the class of planar graphs, then Builder can force a monochromatic copy of a planar graph $G$ if and only if $G$ is outerplanar. Here we show that the "only if" part does not hold while the "if" part does.

On balanced coloring games in random graphs

Consider the balanced Ramsey game, in which a player has r colors and where in each step r random edges of an initially empty graph on n vertices are presented. The player has to immediately assign a different color to each edge and her goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. The Achlioptas game is similar, but the player only loses when she creates a copy of F in one distinguished color. We show that there is an infinite family of non-forests F for which the balanced Ramsey game has a different threshold than the Achlioptas game, settling an open question by Krivelevich et al. We also consider the natural vertex analogues of both games and show that their thresholds coincide for all graphs F, in contrast to our results for the edge case.

A threshold for the Maker‐Breaker clique game

ABSTRACTWe study the Maker‐Breaker k‐clique game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k‐clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win this game is at , for all k > 3, thus proving a conjecture from Ref. [Stojaković and Szabó, Random Struct Algor 26 (2005), 204–223]. More precisely, we conclude that there exist constants such that when the game is Maker's win a.a.s., and when it is Breaker's win a.a.s.For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n, p). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 318–341, 2014

Multicolor on-line degree Ramsey numbers of trees

In the on-line Ramsey game on a family H of graphs, “Builder” presents edges of a graph one-by-one, and “Painter” colors each edge as it is presented; we require that Builder keep the presented graph in H. Builder wins the game (G, H) if Builder can ensure that a monochromatic G arises. The s-color on-line degree Ramsey number of G, denoted ˚ R¢(G; s), is the least k such that Builder wins (G, H) when H is the family of graphs having maximum degree at most k and Painter has s colors available. More generally, ˚ R¢(G1, . . . , Gs) is the minimum k such that Builder can force a copy of Gi in color i for some i when restricted to graphs with maximum degree at most k. In this paper, we prove that ˚ R¢(T ; s) ≤ s(¢( T ) − 1) + 1 for every tree T ; this is sharp, with equality whenever T has adjacent vertices of maximum degree. We also give lower and upper bounds on ˚ R¢(G1, . . . , Gs) when each Gi is a double-star. When each Gi is a star, we determine ˚ R¢(G1, . . . , Gs) exactly.

On Balanced Coloring Games in Random Graphs

Abstract Consider the balanced Ramsey game, in which a player has r colors and where in each round r random edges of an initially empty graph on n vertices are presented. The player has to immediately assign a different color to each edge and her goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. The Achlioptas game is similar, but the player only loses when she creates a copy of F in one distinguished color. We show that there is an infinite family of nonforests F for which the balanced Ramsey game has a different threshold than the Achlioptas game, settling an open question by Krivelevich et al. We also consider the natural vertex analogues of both games and show that their thresholds coincide for all graphs F , in contrast to our results for the edge case.

Ramsey games with giants

AbstractThe classical result in the theory of random graphs, proved by Erdős and Rényi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in a sequence of rounds must be colored with one of r colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011