No abstract

We consider the strong Ramsey-type game R (k) (H, ℵ 0 ), played on the edge set of the infinite complete k-uniform hypergraph K k N . Two players, called FP (the first player) and SP (the second player), take turns claiming edges of K k N with the goal of building a copy of some finite predetermined k-uniform hypergraph H. The first player to build a copy of H wins. If no player has a strategy to ensure his win in finitely many moves, then the game is declared a draw.In this paper, we construct a 5-uniform hypergraph H such that R (5) (H, ℵ 0 ) is a draw. This is in stark contrast to the corresponding finite game R (5) (H, n), played on the edge set of K 5 n . Indeed, using a classical game-theoretic argument known as strategy stealing and a Ramsey-type argument, one can show that for every k-uniform hypergraph G, there exists an integer n 0 such that FP has a winning strategy for R (k) (G, n) for every n ≥ n 0 .

No abstract

No abstract