Non-meager free sets and independent families

Abstract

Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\le \omega$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset $F$ of $X$) such that $F$ is $R$-free for every $R\in \mathcal{R}$. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on $\omega$, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting "$|\mathcal{R}|\le \omega$" with "$|\mathcal{R}|<\mathfrak{c}$" and "Baire" with "completely Baire".

13 pages

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