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arXiv · 2510.07669

Supercompact Measures and the Galvin Property

Tom Benhamou·Ben-Zion Weltsch
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Abstract

We study saturation properties of $\sigma$-complete measures on $P_\kappa(\lambda)$, where $\lambda$ 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 $P_\kappa(\lambda)$, 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 $P_\kappa(\lambda)$ measures. In the second part of the paper, we study variations of the Galvin property suitable for ultrafilters over $P_\kappa(\lambda)$, 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.

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BibTeXRIS

Tom Benhamou, Ben-Zion Weltsch. 2025-10-09. Supercompact Measures and the Galvin Property. https://doi.org/10.1017/jsl.2026.10216

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