Kurepa trees and the failure of the Galvin property

Abstract

We force the existence of a non-trivial κ \kappa -complete ultrafilter over κ \kappa which fails to satisfy the Galvin property. This answers a question asked by Benhamou and Gitik [Ann. Pure Appl. Logic 173 (2022), Paper No. 103107].

On Cohen and Prikry Forcing Notions

“…6, where we start from a single measurable cardinal, and construct a model with an ultrafilter which fails to satisfy the Galvin property. This improves the initial large cardinal assumption of [5].…”

“…In [5], Garti, Shelah, and the first author constructed a model with a -complete ultrafilter which contains and fails to satisfy the Galvin property. The initial assumption was a supercompact cardinal and the construction went through adding slim Kurepa trees.…”

“…In [1], Abraham and Shelah constructed a model where Gal (Cub κ + , κ + , κ ++ ) fails for a regular κ. Garti [13,14] and later together with the first author and Poveda [4] continued the investigation of the Galvin property for the club filter. The Galvin property for κ-complete ultrafilters over a measurable cardinal κ was used recently in [7,18].…”

Negating the Galvin property

“…On a similar vein, the large-cardinal hypothesis used in Theorem 2.3 has been considerably weakened -the same configuration is forceable just from a (𝜅 + 3)-strong cardinal 𝜅. Question 5.10 was also settled in the affirmative in [8,Theorem 3.2].…”

“…In recent times, Galvin's property has experienced a renewed interest after finding deep connections with the structure of Prikry-type generic extensions [10,33]. In a different direction, some other interesting combinatorial implications of this principle have been discovered in the area of polarized relations [9]. In fact, the authors endorse the thesis that a deeper understanding of Galvin's property -and some of the variants considered in this paper -may led to open new directions in set theory and other related areas.…”