Kurepa trees and the failure of the Galvin property

TL;DR

This paper demonstrates, through forcing, the existence of a non-trivial ultrafilter over a cardinal that does not satisfy the Galvin property, addressing an open question in set theory.

Contribution

It introduces a forcing method to produce a $ ext{kappa}$-complete ultrafilter that fails the Galvin property, providing a counterexample to a previously open question.

Findings

Existence of a $ ext{kappa}$-complete ultrafilter failing the Galvin property

Forcing technique to construct such ultrafilters

Answers an open question by the authors and Gitik

Abstract

We force the existence of a non-trivial $\kappa$-complete ultrafilter over $\kappa$ which fails to satisfy the Galvin property. This answers a question asked by the first author and Moti Gitik.