Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

TL;DR

This paper uses topological Ramsey space methods to compute the Ramsey degrees of various ultrafilters generated by sigma-closed forcings and explores their pseudointersection and tower numbers, linking these to classical cardinal invariants.

Contribution

It introduces a unified approach to calculating Ramsey degrees of ultrafilters from sigma-closed forcings and extends known results with new calculations and streamlined proofs.

Findings

Calculated Ramsey degrees for several classes of ultrafilters.

Established relationships between pseudointersection/tower numbers and classical invariants.

Provided new and simplified proofs for existing results.

Abstract

This paper investigates properties of $\sigma$-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter $\mathcal{U}$ for $n$-tuples, denoted $t(\mathcal{U},n)$, is the smallest number $t$ such that given any $l\ge 2$ and coloring $c:[\omega]^n\rightarrow l$, there is a member $X\in\mathcal{U}$ such that the restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known $\sigma$-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by $\sigma$-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings $\mathcal{P}(\omega^k)/\mathrm{Fin}^{\otimes k}$. We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these $\sigma$-closed forcings and their relationships with the classical pseudointersection number $\mathfrak{p}$.