On filling families of 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.