The Galvin property under the Ultrapower Axiom

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 $\hat{I}º$-complete ultrafilter has the Galvin property if and only if it is an iterated sum of $p$-points.

Added references and attributions, reorganized the proof of the main theorem, no new ideas

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• The Galvin property under the Ultrapower Axiom

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• The Galvin property under the Ultrapower Axiom