For two graphs B and H the strong Ramsey game R(B,H) on the board B and with target H is played as follows. Two players alternately claim edges of B. The first player to build a copy of H wins. If none of the players win, the game is declared a draw. A notorious open question of Beck (1996, 2002, 2011) asks whether the first player has a winning strategy in R(Kn,Kk) in bounded time as n→∞. Surprisingly, in a recent paper Hefetz et al. (2017) constructed a 5-uniform hypergraph H for which they proved that the first player does not have a winning strategy in R(Kn(5),H) in bounded time. They naturally ask whether the same result holds for graphs. In this paper we make further progress in decreasing the rank.
In our first result, we construct a graph G (in fact G=K6∖K4) and prove that the first player does not have a winning strategy in R(Kn⊔Kn,G) in bounded time. As an application of this result we deduce our second result in which we construct a 4-uniform hypergraph G′ and prove that the first player does not have a winning strategy in R(Kn(4),G′) in bounded time. This improves the result in the paper above.
By compactness, an equivalent formulation of our first result is that the game R(Kω⊔Kω,G) is a draw. Another reason for interest on the board Kω⊔Kω is a folklore result that the disjoint union of two finite positional games both of which are first player wins is also a first player win. An amusing corollary of our first result is that at least one of the following two natural statements is false: (1) for every graph H, R(Kω,H) is a first player win; (2) for every graph H if R(Kω,H) is a first player win, then R(Kω⊔Kω,H) is also a first player win. Surprisingly, we cannot decide between the two.