The Galvin Property Under the Ultrapower Axiom

Abstract

We continue the study of the Galvin property from [6] and [1]. In particular, we deepen the connection between certain diamondlike 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 [2]. Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a κ-complete ultrafilter has the Galvin property if and only if it is an iterated sum of p-points.