Hefetz, D;
Kusch, C;
Narins, L;
Pokrovskiy, A;
Requile, C;
Sarid, A;
(2017)
Strong Ramsey games: Drawing on an infinite board.
Journal of Combinatorial Theory, Series A
, 150
pp. 248-266.
10.1016/j.jcta.2017.03.008.
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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 K5 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 K5, 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 lead most people (including the authors) to believe that the K5 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.
| Type: | Article |
|---|---|
| Title: | Strong Ramsey games: Drawing on an infinite board |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.jcta.2017.03.008 |
| Publisher version: | https://doi.org/10.1016/j.jcta.2017.03.008 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Positional games, Strong games, Ramsey Theory |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10112657 |
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