Geodesic
The Galvin property under the ultrapower axiom
Canadian journal of mathematics, Tome 77 (2025) no. 5, pp. 1686-1717

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DOI

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.
DOI : 10.4153/S0008414X2400052X
Mots-clés : Galvin’s property, the Ultrapower Axiom, inner models, the Tukey order, p-point ultrafilters 
Benhamou, Tom; Goldberg, Gabriel. The Galvin property under the ultrapower axiom. Canadian journal of mathematics, Tome 77 (2025) no. 5, pp. 1686-1717. doi: 10.4153/S0008414X2400052X
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