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For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on theboard $B$ and with target $H$ is played as follows. Two players alternatelyclaim edges of $B$. The first player to build a copy of $H$ wins. If none ofthe players win, the game is declared a draw. A notorious open question of Beckasks whether the first player has a winning strategy in $\mathcal{R}(K_n,K_k)$in bounded time as $n\rightarrow\infty$. Surprisingly, in a recent paper Hefetzet al. constructed a $5$-uniform hypergraph $\mathcal{H}$ for which they provedthat the first player does not have a winning strategy in$\mathcal{R}(K_n^{(5)},\mathcal{H})$ in bounded time. They naturally askwhether the same result holds for graphs. In this paper we make furtherprogress in decreasing the rank.  In our first result, we construct a graph $G$ (in fact $G=K_6\setminus K_4$)and prove that the first player does not have a winning strategy in$\mathcal{R}(K_n \sqcup K_n,G)$ in bounded time. As an application of thisresult we deduce our second result in which we construct a $4$-uniformhypergraph $G'$ and prove that the first player does not have a winningstrategy in $\mathcal{R}(K_n^{(4)},G')$ in bounded time. This improves theresult in the paper above.  An equivalent formulation of our first result is that the game$\mathcal{R}(K_\omega\sqcup K_\omega,G)$ is a draw. Another reason for intereston the board $K_\omega\sqcup K_\omega$ is a folklore result that the disjointunion of two finite positional games both of which are first player wins isalso a first player win. An amusing corollary of our first result is that atleast one of the following two natural statements is false: (1) for every graph$H$, $\mathcal{R}(K_\omega,H)$ is a first player win; (2) for every graph $H$if $\mathcal{R}(K_\omega,H)$ is a first player win, then$\mathcal{R}(K_\omega\sqcup K_\omega,H)$ is also a first player win.

Strong Ramsey Games in Unbounded Time