Supercompact Measures and the Galvin Property
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915541376040960 |
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| author | Benhamou, Tom Weltsch, Ben-Zion |
| author_facet | Benhamou, Tom Weltsch, Ben-Zion |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Supercompact Measures and the Galvin Property Benhamou, Tom Weltsch, Ben-Zion Logic 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. |
| title | Supercompact Measures and the Galvin Property |
| topic | Logic |
| url | https://arxiv.org/abs/2510.07669 |