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 $|\mathcal {R}|\leq \omega$, 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 “$|\mathcal {R}|\leq \omega$” with “$|\mathcal {R}|<\mathfrak {c}$” and “Baire” with “completely Baire”.