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

TL;DR

This paper constructs a specific ultrafilter over a cardinal  that demonstrates the failure of the Galvin property at ^+ but not at ^{++}, answering a question posed by previous researchers.

Contribution

It provides a novel construction of a -complete ultrafilter with selective Galvin property failures at different cardinal levels.

Findings

Constructed a -complete ultrafilter with specific Galvin property behavior.

Showed the Galvin property can fail at ^+ but hold at ^{++}.

Answered an open question by Benhamou and Goldberg.

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.