Strong Ramsey Games: Drawing on an infinite board

We consider the strong Ramsey-type game $$\mathcal{R}$^{(k)}($\mathcal{H}$, \aleph_0)$,playedontheedgesetoftheinfinitecomplete$k$-uniform hypergraph $K_{\mathbb{N}}^k$. Two players, called FP (the first player) and SP (the second player), take turns claiming edges of $K_{\mathbb{N}}^k$ with the goal of building a copy of some finite predetermined $k$-uniform hypergraph $\mathcal{H}$. The first player to build a copy of $\mathcal{H}$ 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 $\mathcal{H}$ such that ${\mathcal{R}}^{(5)}(\mathcal{H},{\mathrm{\aleph }}_0)$ is a draw. This is in stark contrast to the corresponding finite game ${\mathcal{R}}^{(5)}(\mathcal{H},n)$, played on the edge set of $K_n^5$. Indeed, using a classical game-theoretic argument known as \emph{strategy stealing} and a Ramsey-type argument, one can show that for every $k$-uniform hypergraph $\mathcal{G}$, there exists an integer $n_0$ such that FP has a winning strategy for ${\mathcal{R}}^{(k)}(\mathcal{G},n)$ for every $n\ge n_0$.