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

TL;DR

This work investigates Galvin's property for $\kappa$-complete ultrafilters on very large cardinals and its impact on partition calculus. It develops forcing strategies, notably Generalized Mathias forcing and Gitik–Shelah iterations, to control whether ground-model $\kappa$-complete ultrafilters extend to non-Galvin, Galvin, or $P$-point ultrafilters, while preserving large-cardinal features such as supercompactness, $C^{(n)}$-extendibility, and VP. In parallel, it shows that Galvin-like behavior can persist in choiceless settings, with results under boldface $GCH$ and under AD that yield definable Galvin properties for many cardinals. Finally, these tools are applied to partition calculus, demonstrating new instances of $\lambda\rightarrow(\lambda,\omega+1)^2$ and clarifying the relationship between Galvin's property and classical partition relations across both ZFC and ZF frameworks.

Abstract

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 $λ\rightarrow(λ,ω+1)^2$.

Theorems & Definitions (63)

• Proposition 2.1
• proof
• Claim 2.2
• proof : Proof of claim
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5
• proof