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

Abstract

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