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

In the first part of this paper we produce models where κ is certain very large cardinal and every ground model κ-complete ultrafilter extends to a non-Galvin one. In the opposite direction, we also produce such models but this time every ground model κ-complete ultrafilter extends to a P-point ultrafilter, hence to a Galvin one. Finally, we apply Galvin’s property to obtain consistently new instances of the classical problem in partition calculus λ → (λ, ω + 1)² both in ZFC and ZF.