Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hrušák
Andrea Medini
TL;DR
This work investigates when products of filters on $\omega$ are countable dense homogeneous ($\mathsf{CDH}$) and reveals a sharp link to combinatorial cardinals. It proves a partial answer to Hrušák's question: if $\prod_{\alpha\in\kappa}\mathcal F_\alpha$ is $\mathsf{CDH}$, then $\kappa<\mathfrak{p}$ and each factor is a non-meager $\mathsf{P}$-filter, and shows that the product of fewer than $\mathfrak{p}$ such filters has the Miller property ($\mathsf{MP}$). The paper also clarifies the relationship between Miller-measurability and the Miller property, establishing that intersections of fewer than $\mathrm{add}(m^0)$ non-meager $\mathsf{P}$-filters remain non-meager $\mathsf{P}$-filters, where $m^0$ is the Miller ideal. Finally, it develops preservation results: the Miller property is preserved under certain intersections and, more broadly, under countable products, linking topological and set-theoretic aspects of filters and expanding the toolbox for analyzing $\mathsf{CDH}$ phenomena.
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.