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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(K-n, K-k) in bounded time as n -> infinity. 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(K-n((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 = K-6\\K-4) and prove that the first player does not have a winning strategy in R(K-n boolean OR K-n, 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(K-n((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(K-omega boolean OR K-omega, G) is a draw. Another reason for interest on the board K-omega boolean OR K-omega 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(K-omega, H) is a first player win; (2) for every graph H if R(K-omega, H) is a first player win, then R(K-omega boolean OR K-omega, H) is also a first player win. Surprisingly, we cannot decide between the two. (C) 2020 Elsevier Ltd. 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