The Galvin property under the ultrapower axiom

We continue the study of the Galvin property from Benhamou, Garti, and Shelah (2023, Proceedings of the American Mathematical Society 151, 1301-1309) and Benhamou (2023, Saturation properties in canonical inner models, submitted). 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 Benhamou and Dobrinen (2023, Journal of Symbolic Logic, 1-34). 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.