Seven characterizations of non-meager P-filters

Abstract

We give several topological/combinatorial conditions that, for a filter on $ω$, are equivalent to being a non-meager $\mathsf{P}$-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager $\mathsf{P}$-filter. Here, we identify a filter with a subspace of $2^ω$ through characteristic functions. Along the way, we generalize to non-meager $\mathsf{P}$-filters a result of Miller about $\mathsf{P}$-points, and we employ and give a new proof of results of Marciszewski. We also employ a theorem of Hernández-Gutiérrez and Hrušák, and answer two questions that they posed. Our result also resolves several issues raised by Medini and Milovich, and proves false one "theorem" of theirs. Furthermore, we show that the statement "Every non-meager filter contains a non-meager $\mathsf{P}$-subfilter" is independent of $\mathsf{ZFC}$ (more precisely, it is a consequence of $\mathfrak{u}<\mathfrak{g}$ and its negation is a consequence of $\Diamond$). It follows from results of Hrušák and van Mill that, under $\mathfrak{u}<\mathfrak{g}$, a filter has less than $\mathfrak{c}$ types of countable dense subsets if and only if it is a non-meager $\mathsf{P}$-filter. In particular, under $\mathfrak{u}<\mathfrak{g}$, there exists an ultrafilter with $\mathfrak{c}$ types of countable dense subsets. We also show that such an ultrafilter exists under $\mathsf{MA(countable)}$.