Strong Ramsey games: Drawing on an infinite board

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.