Supercompact Measures and the Galvin Property

2025/10/09 by Benhamou, Tom, Weltsch, Ben-Zion
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2510.07669

Abstract

We study saturation properties of σ-complete measures on P_κ(λ), where λ can be either regular or singular. In particular, we prove that in contrast to Galvin's theorem, the Galvin property of Benhamou-Garti-Poveda fails for normal fine ultrafilters on P_κ(λ), answering a question of the first author and Goldberg. We then provide several applications of our results: to ultrafilters on successor cardinals under UA, we generalize a result of Gitik regarding density of ground model sets in supercompact Prikry extensions, and to generating sets of P_κ(λ) measures. In the second part of the paper, we study variations of the Galvin property suitable for ultrafilters over P_κ(λ), and generalize a result of Foreman-Magidor-Zeman on determinacy of filter games to the two-cardinal setting, answering a question of the first author and Gitman.

Citations

• Ultrafilters over Successor Cardinals and the Tukey Order
• Dense ideals
• On Ultrapowers and Cohesive Ultrafilters
• Diamond principles and Tukey-top ultrafilters on a countable set
• The Galvin property under the Ultrapower Axiom
• Games with Filters
• The Ultrapower Axiom
• A small ultrafilter number at smaller cardinals
• Tukey types of ultrafilters

Related

• Saturation Properties of Ultrafilters in Canonical Inner Models
• Galvin's property at large cardinals and an application to partition calculus
• The Galvin property under the Ultrapower Axiom
• Measures that violate the Generalized Continuum Hypothesis
• The Galvin property at κ⁺⁺ and not at κ⁺