  We continue the study of the Galvin property. In particular, we deepen the
connection between certain diamond-like principles and non-Galvin ultrafilters.
We also show that any Dodd sound ultrafilter that is not a $p$-point is
non-Galvin. We use these ideas to formulate an essentially optimal large
cardinal hypothesis that ensures the existence of a non-Galvin ultrafilter,
improving on results of Benhamou and Dobrinen. Finally, we use a strengthening
of the Ultrapower Axiom to prove that in all the known canonical inner models,
a $\kappa$-complete ultrafilter on $\kappa$ has the Galvin property if and only
if it is an iterated sum of $p$-points.
