The Galvin property under the Ultrapower Axiom

TL;DR

This paper investigates the conditions under which ultrafilters possess the Galvin property, exploring connections with diamond principles, large cardinals, and inner models, and establishing new characterizations and hypotheses.

Contribution

It deepens the understanding of the Galvin property in ultrafilters, links it with diamond-like principles, and identifies optimal large cardinal hypotheses for non-Galvin ultrafilters.

Findings

Dodd sound non p-point ultrafilters are non-Galvin.

Formulation of an optimal large cardinal hypothesis for non-Galvin ultrafilters.

In canonical inner models, a κ-complete ultrafilter has the Galvin property iff it is an iterated sum of p-points.

Abstract

We continue the study of the Galvin property from \cite{bgs} and \cite{Benhamou2}. In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound non p-point ultrafilter is non-Galvin. We use these ideas to formulate what appears to be the optimal large cardinal hypothesis implying the existence of a non-Galvin ultrafilter, improving on a result from \cite{Benhamou_Dobrinen}. Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a $\kappa$-complete ultrafilter has the Galvin property if and only if it is an iterated sum of $p$-points.