On filling families of finite subsets of the Cantor set

TL;DR

This paper investigates conditions under which certain hereditary families of finite subsets of the Cantor set contain perfect subsets, extending previous results by considering measure and density conditions.

Contribution

It establishes that $ ext{C}$-measurable $ ext{ee}$-filling families over the Cantor set contain perfect subsets, generalizing earlier findings to measure and density contexts.

Findings

Existence of perfect subsets within $ ext{C}$-measurable $ ext{ee}$-filling families.

Extension of results to families with weaker density conditions.

Generalization of previous theorems on finite subsets of the Cantor set.

Abstract

Let $\ee>0$ and $\fff$ be a family of finite subsets of the Cantor set $\ccc$. Following D. H. Fremlin, we say that $\fff$ is $\ee$-filling over $\ccc$ if $\fff$ is hereditary and for every $F\subseteq\ccc$ finite there exists $G\subseteq F$ such that $G\in\fff$ and $|G|\geq\ee |F|$. We show that if $\fff$ is $\ee$-filling over $\ccc$ and $C$-measurable in $[\ccc]^{<\omega}$, then for every $P\subseteq\ccc$ perfect there exists $Q\subseteq P$ perfect with $[Q]^{<\omega}\subseteq\fff$. A similar result for weaker versions of density is also obtained.