A unique $Q$-point and infinitely many near-coherence classes of ultrafilters

TL;DR

This paper demonstrates that in a specific set-theoretic model, there can be a unique $Q$-point ultrafilter up to isomorphism while simultaneously having a continuum of near-coherence classes of ultrafilters, highlighting the complex structure of ultrafilters.

Contribution

It constructs a model where exactly one $Q$-point exists up to isomorphism, yet there are continuum many near-coherence classes, showing the independence of these properties.

Findings

Existence of a unique $Q$-point ultrafilter in the model.

Presence of continuum many near-coherence classes of ultrafilters.

Consistency results related to ultrafilter structures.

Abstract

We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length-$\omega_2$ countable support iteration of restricted Mathias forcing over a ground model satisfying $\textsf{CH}$, there is a unique $Q$-point up to isomorphism. In particular, it is consistent that there is only one $Q$-point while there are $2^{\mathfrak{c}}$-many near-coherence classes of ultrafilters.