Here, we identify subsets of $\mathbb{N}$ with sequences obtained by listing the members of the set in strictly increasing order. Suppose that we have some set $\mathcal{F}$ of sets (sequences) of the form $(m_n, l_n)_{n=1}^t$. We let $\mathcal{U}$ denote the set of all $(m_n, l_n)_{n=1}^\infty$ such that $(m_n, l_n)_{n=1}^t\in \mathcal{F}$ for all $t\in\mathbb{N}$. We let $\mathcal{V}$ denote the set of all $(m_n)_{n=1}^\infty$ such that there exists $(l_n)_{n=1}^\infty$ such that $(m_n, l_n)_{n=1}^\infty \in \mathcal{U}$.

It is easy to see that $\mathcal{U}$ is closed and $\mathcal{V}$ is analytic (in either the Cantor or Ellentuck topology). Suppose also that for every infinite subset $K$ of $\mathbb{N}$, there exists a further, infinite subset $M$ of $L$ such that every infinite subset of $M$ lies in $\mathcal{V}$.

Is it known that the latter property is equivalent to a winning strategy for a certain player in some two player game?