Strong Ramsey games: Drawing on an infinite board

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:150

Article

Hefetz, Dan¹  Kusch, Christopher^2,3  Narins, Lothar^2,3  Pokrovskiy, Alexey⁴  Requile, Clement^2,3 

[1] Ariel Univ, Dept Comp Sci, IL-40700 Ariel, Israel

[2] Free Univ Berlin, Inst Math & Informat, Berlin, Germany

[3] Berlin Math Sch, Berlin, Germany

[4] Tel Aviv Univ, Sch Math Sci, Raymond & Beverly Sackler Fac Exact Sci, IL-6997801 Tel Aviv, Israel

关键词: Positional games;    Strong games;    Ramsey Theory;   

DOI  :  10.1016/j.jcta.2017.03.008

来源: Elsevier

【 摘 要 】

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 K-q (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 K-n 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 K-q: 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 K-q 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. (C) 2017 Published by Elsevier Inc.

【 授权许可】

Free