On filling families of finite subsets of the Cantor set

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