Local Ramsey Spaces in Matet Forcing Extensions and Finitely Many Near-Coherence Classes

TL;DR

This paper introduces a new forcing method using Gowers--Matet forcing with multiple ultrafilters to precisely control the number of near-coherence classes of ultrafilters in extensions, solving a longstanding problem.

Contribution

It develops a novel forcing technique with multiple ultrafilters to determine the exact number of near-coherence classes in extensions, advancing ultrafilter classification.

Findings

For any finite n ≥ 1, there exists a forcing extension with exactly n near-coherence classes.

The paper proves a strengthened version of Gowers's theorem on colorings of Fin_k.

The new forcing method settles the spectrum of near-coherence classes in set-theoretic extensions.

Abstract

We introduce Gowers--Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over $\omega$, and with this forcing we settle the long-standing problem of the spectrum of numbers near-coherence classes. We prove that for any finite $n \geq 1$, there is a forcing extension with exactly $n$ near-coherence classes of ultrafilters. For evaluating the new forcing, we prove a strengthening of Gowers's theorem on colourings of ${\rm Fin}_k$.