Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hru\v{s}\'{a}k

Abstract

Given a cardinal $\kappa$ and filters $\mathcal{F}_\alpha$ on $\omega$ for $\alpha\in\kappa$, we will show that if $\prod_{\alpha\in\kappa}\mathcal{F}_\alpha$ is countable dense homogeneous then $\kappa<\mathfrak{p}$ and each $\mathcal{F}_\alpha$ is a non-meager $\mathsf{P}$-filter. This partially answers a question of Michael Hru\v{s}\'{a}k. 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.