The Galvin property at ${\kappa }^{++}$ and not at ${\kappa }^{+}$

Abstract

We construct a $\kappa -$complete ultrafilter $W$ over $\kappa$ such that $\neg$Gal$(\kappa ,W,{\kappa }^{+})$ and Gal$(\kappa ,W,{\kappa }^{++})$. This answers a question of T. Benhamou and G. Goldberg.