Free sets for a set-mapping relative to a family of sets

Abstract

Given a family $\mathcal{F}$ of subsets of $\{1,\ldots,m\}$, we try to compute the least natural number $n$ such that for every function $S:[\aleph_n]^{<\omega}\longrightarrow [\aleph_n]^{<\omega}$ there exists a bijection $u:\{1,\ldots,m\}\longrightarrow Y\subset \aleph_n$ such that $Su(A)\cap Y \subset u(A)$ for all $A\in\mathcal{F}$.