Saturation Properties of Ultrafilters in Canonical Inner Models

TL;DR

This paper investigates the saturation properties of ultrafilters in canonical inner models, showing that certain models satisfy the Galvin property while supercompact cardinals do not, and linking these properties to combinatorial principles.

Contribution

It extends Galvin's Theorem to p-point limits of p-points and characterizes ultrafilters in inner models and large cardinal contexts regarding the Galvin property.

Findings

Canonical inner models up to superstrong cardinals satisfy the Galvin property for ultrafilters.

Supercompact cardinals always carry non-Galvin ultrafilters.

The principle iamondsuit(ppa) implies the existence of non-Galvin ppa-complete filters.

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}.