Galvin's property at large cardinals and an application to partition calculus
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866911250622971904 |
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| author | Benhamou, Tom Garti, Shimon Poveda, Alejandro |
| author_facet | Benhamou, Tom Garti, Shimon Poveda, Alejandro |
| contents | In the first part of this paper, we explore the possibility for a very large cardinal $κ$ to carry a $κ$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $κ$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $κ$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. Finally, we apply this property to obtain consistently new instances of the classical problem in partition calculus $λ\rightarrow(λ,ω+1)^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_07401 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Galvin's property at large cardinals and an application to partition calculus Benhamou, Tom Garti, Shimon Poveda, Alejandro Logic 03E02, 03E35, 03E60 In the first part of this paper, we explore the possibility for a very large cardinal $κ$ to carry a $κ$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $κ$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $κ$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. Finally, we apply this property to obtain consistently new instances of the classical problem in partition calculus $λ\rightarrow(λ,ω+1)^2$. |
| title | Galvin's property at large cardinals and an application to partition calculus |
| topic | Logic 03E02, 03E35, 03E60 |
| url | https://arxiv.org/abs/2207.07401 |