Related papers: The Galvin property under the Ultrapower Axiom

We improve Galvin's Theorem for ultrafilters which are p-point limits of p-points. This implies that in all the canonical inner models up to a superstrong cardinal, every $\kappa$-complete ultrafilter over a measurable cardinal $\kappa$…

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…

We study saturation properties of $\sigma$-complete measures on $P_\kappa(\lambda)$, where $\lambda$ can be either regular or singular. In particular, we prove that in contrast to Galvin's theorem, the Galvin property of…

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.

We address the question of the consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions \cite[Questions 7.8,7.9]{TomMotiII}, \cite[Question 5]{NegGalSing} and improve theorem…

We develop the theory of cofinal types of ultrafilters over measurable cardinals and establish its connections to Galvin's property. We generalize fundamental results from the countable to the uncountable, but often in surprisingly…

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's…

We characterize the Tukey order, the Galvin property/ Cohesive ultrafilters from \cite{Kanamori1978} in terms of ultrapowers. We use this characterization to measure the distance between the Tukey order and other well-known orders of…

We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$-complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$. This…

We provide two types of guessing principles for ultrafilter ($\diamondsuit^{-}_{\lambda}(U), \ \diamondsuit^p_\lambda(U)$) on $\omega$ which form subclasses of Tukey-top ultrafilters, and construct such ultrafilters in $ZFC$. These…

We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.

We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower…

We characterize strong $p$-point ultrafilters by showing that they are exactly those $p$-points that are not Tukey above $(\omega^\omega,\leq)$; or equivalently, those $p$-points that are not Tukey-idempotent. Moreover, we show that there…

Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we…

We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the…

All ultrafilters under consideration here are non-principal ultrafilters on the set omega of natural numbers. We are concerned with the possible cofinalities of ultrapowers of omega with respect to such ultrafilters. We show that no…

A simple \(P_\lambda\)-point on a regular cardinal \(\kappa\) is a uniform ultrafilter on \(\kappa\) with a mod-bounded decreasing generating sequence of length \(\lambda\). We prove that if there is a simple $P_\lambda$-point ultrafilter…

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin's Axiom for countable posets, to distinguish non-principal ultrafilters on $\omega$ up to homeomorphism.…

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.

We investigate the question of whether $\mathbb Q$ carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this…