Strong Ramsey games: Drawing on an infinite board

Dan Hefetz, Christopher Kusch, Lothar Narins, Alexey Pokrovskiy, Clément Requilé, Amir Sarid

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)248-266
Number of pages19
JournalJournal of Combinatorial Theory. Series A
Volume150
DOIs
StatePublished - 1 Aug 2017

Keywords

  • Positional games
  • Ramsey Theory
  • Strong games

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