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
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. 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.
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.Comment: 17 pages, 48 figures; improved presentation, particularly in section