Galvin's property at large cardinals and an application to partition calculus

TL;DR

This paper investigates the behavior of large cardinal ultrafilters with respect to Galvin's property, demonstrating consistency results and applying findings to partition calculus problems.

Contribution

It establishes the consistency of ultrafilters extending to non-Galvin or P-point ultrafilters and applies these results to classical partition calculus problems.

Findings

Consistency of ultrafilters extending to non-Galvin ultrafilters

Consistency of ultrafilters extending to P-point ultrafilters

New instances of partition calculus results

Abstract

In the first part of this paper, we explore the possibility for a very large cardinal $\kappa$ to carry a $\kappa$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $\kappa$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $\kappa$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. Finally, we apply this property to obtain consistently new instances of the classical problem in partition calculus $\lambda\rightarrow(\lambda,\omega+1)^2$.