Strong Ramsey games: Drawing on an infinite board

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 .

Strong Ramsey games in unbounded time

“…These examples show that a good strong game strategy should by all means seek to interact with the other player. However, the main approach to strong games so far relies on finding fast "Maker" strategies for the corresponding weak game and transforming them into strategies for the strong games [11,12,16]. Thus, our strategy suggests that the Ramsey game requires more sophisticated arguments.…”

“…However, the corresponding question for graphs, as asked in [16], still remains open. As we explain in Section 1.3, as the rank decreases this question becomes much harder.…”

“…In our first result, Theorem 1.2 (see Corollary 1.3), we exhibit a graph G (in fact G = K 6 \K 4 , see Figure 1), for which we prove that in R(K n ⊔K n , G) P1 does not have a winning strategy in bounded time. In order to do so we build on the work of [16]. However, there is a serious obstacle in adapting the strategy developed in [16] from 5-uniform hypergraphs to graphs.…”

“…In order to do so we build on the work of [16]. However, there is a serious obstacle in adapting the strategy developed in [16] from 5-uniform hypergraphs to graphs. As we further explain in Section 1.3, although the strategy developed in [16] is a strong game drawing strategy, the core of the argument is a weak game fast winning strategy that is inapplicable to graphs.…”

“…In the opposite direction, another notorious open question asks if for every fixed target graph H in the game R(K n , H), P1 has a winning strategy in bounded time as n → ∞ or, equivalently, R(K ω , H) is a P1 win. In a recent paper [16], Hefetz, Kusch, Narins, Pokrovskiy, Requilé and Sarid addressed the natural generalisation to hypergraphs, and changed the intuition about this phenomenon completely. Surprisingly, in [16] they exhibited a target 5-uniform hypergraph H for which they constructed an explicit drawing strategy for P2 in the game R K (5) ω , H .…”