Stable ordered union ultrafilters and $\mathrm{cov}(\mathcal{M})<\mathfrak c$

TL;DR

This paper constructs models of ZFC where stable ordered union ultrafilters exist despite the covering number of the meager ideal being less than the continuum, challenging previous assumptions about their relationship.

Contribution

It demonstrates the consistency of union ultrafilters with the inequality ( ext{meager})< ext{continuum} within ZFC models, expanding understanding of ultrafilter existence.

Findings

Models of ZFC with union ultrafilters and ( ext{meager})< ext{continuum}

Ultrafilter existence independent of ( ext{meager})= ext{continuum}

New techniques for constructing such models

Abstract

A union ultrafilter is an ultrafilter over the finite subsets of $\omega$ that has a base of sets of the form $\mathrm{FU}(X)$, where $X$ is an infinite pairwise disjoint family and $\mathrm{FU}(X)=\{\bigcup F\big|F\in[X]^{<\omega}\setminus\{\varnothing\}\}$. The existence of these ultrafilters is not provable from the $\mathsf{ZFC}$ axioms, but is known to follow from the assumption that $\mathrm{cov}(\mathcal{M})=\mathfrak c$. In this article we obtain various models of $\mathsf{ZFC}$ that satisfy the existence of union ultrafilters while at the same time $\mathrm{cov}(\mathcal{M})<\mathfrak c$.