Saturation Properties of Ultrafilters in Canonical Inner Models

Tom Benhamou

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arXiv · 2212.14096

Saturation Properties of Ultrafilters in Canonical Inner Models

Tom Benhamou

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Abstract

We improve Galvin's Theorem for ultrafilters which are p-point limits of p-points. This implies that in all the canonical inner models up to a superstrong cardinal, every $\kappa$-complete ultrafilter over a measurable cardinal $\kappa$ satisfies the Galvin property. On the other hand, we prove that supercompact cardinals always carry non-Galvin $\kappa$-complete ultrafilters. Finally, we prove that $\diamondsuit(\kappa)$ implies the existence of a $\kappa$-complete filter which extends the club filter and fails to satisfy the Galvin property. This answers questions \cite[Question 5.22]{TomMotiII},\cite[Question 3.4]{Non-GalvinFil} and questions ,\cite[Question 4.5]{BenGarShe},\cite[Question 2.26]{bgp}.

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Tom Benhamou. 2022-12-28. Saturation Properties of Ultrafilters in Canonical Inner Models. https://arxiv.org/abs/2212.14096

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