Supercompact Measures and the Galvin Property
Tom Benhamou, Ben-Zion Weltsch
TL;DR
This work analyzes saturation and regularity phenomena for $\sigma$-complete measures on $P_\kappa(\lambda)$, with a focus on normal, fine ultrafilters in the two-cardinal setting. By organizing cases according to the cofinality of $\lambda$ and leveraging ultrapower techniques, it establishes broad failures of the Galvin property, while isolating a positive regime when ${\rm cf}(\kappa)=\lambda$ (and under $2^{<\lambda}=\lambda$) in which $\mathrm{Gal}(U,\kappa,\lambda^+)$ (and, via RK-equivalence, $\mathrm{Gal}(U,\kappa,2^{\lambda})$) holds. The paper derives several applications: under the Ultrapower Axiom, density results for successor-cardinal ultrafilters; density-of-old-sets results in supercompact Prikry extensions; and structural insights into generating sets for $P_\kappa(\lambda)$-measures. It further develops a revised Galvin property modulo the fine filter and, via two-cardinal filter games, constructs normal, precipitous, $\lambda$-measuring ideals with dense trees, extending the landscape of two-cardinal large-cardinal phenomena and their forcing-theoretic consequences. Overall, the results delineate sharp boundaries for Galvin-type regularity in the two-cardinal setting and provide new tools for understanding Tukey-topness and forcing constructions related to ultrafilters on $P_\kappa(\lambda)$.
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.