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arXiv · 2207.07401

Galvin's property at large cardinals and an application to partition calculus

Tom Benhamou·Shimon Garti·Alejandro Poveda
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Abstract

In the first part of this paper, we explore the possibility for a very large cardinal $\kappa$ to carry a $\kappa$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $\kappa$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $\kappa$-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 $\lambda\rightarrow(\lambda,\omega+1)^2$.

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Tom Benhamou, Shimon Garti, Alejandro Poveda. 2022-07-15. Galvin's property at large cardinals and an application to partition calculus. https://doi.org/10.1007/s11856-025-2749-7

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