Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hrušák

Abstract

Given a cardinal $κ$ and filters $\mathcal{F}_α$ on $ω$ for $α\inκ$, we will show that if $\prod_{α\inκ}\mathcal{F}_α$ is countable dense homogeneous then $κ<\mathfrak{p}$ and each $\mathcal{F}_α$ is a non-meager $\mathsf{P}$-filter. This partially answers a question of Michael Hrušák. Along the way, we will show that the product of fewer than $\mathfrak{p}$ non-meager $\mathsf{P}$-filters has the Miller property. We will also describe explicitly the connection between Miller-measurability and the Miller property. As a corollary, we will see that the intersection of fewer than $\mathsf{add}(m^0)$ non-meager $\mathsf{P}$-filters is a non-meager $\mathsf{P}$-filter, where $m^0$ denotes the ideal of Miller-null sets. We will conclude by investigating the preservation of the Miller property under intersections and products.