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

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 $\lambda \to (\lambda ,\omega +1{)}^2$.

Revisited

References in corpus (4)

• Intermediate Models of Magidor-Radin Forcing- Part II
• Negating the Galvin Property
• Kurepa trees and the failure of the Galvin property
• On Cohen and Prikry Forcing Notions