  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
$\lambda\rightarrow(\lambda,\omega+1)^2$.
