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

TL;DR

This paper investigates the minimal size of a set needed to guarantee the existence of a certain bijection satisfying a set-mapping condition relative to a family of subsets, within a combinatorial set theory framework.

Contribution

It introduces a new combinatorial problem involving set-mappings and families of subsets, providing bounds and conditions for the existence of such bijections.

Findings

Established bounds on the minimal natural number n for given families

Characterized conditions under which the set-mapping property holds

Connected the problem to existing combinatorial set theory concepts

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 $A\in\mathcal{F}$.