Related papers: Monotonicity of the ultrafilter number function

We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located…

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many…

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of…

We extend a theorem by Juh\'asz and Szentmikl\'ossy to notions related to pseudocompactness. We also allow the case when one of the cardinals under consideration is singular. We give an application to the study of decomposable ultrafilters:…

We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly…

We prove that on the Baire space $(D^{\kappa},\pi)$, $\kappa \geq \omega_0$ where $D$ is a uniformly discrete space having $\omega _1$-strongly compact cardinal and $\pi$ denotes the product uniformity on $D^\kappa$, there exists a…

We study ultrafilters on regular uncountable cardinals, with a primary focus on $\omega_1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform…

We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(\kappa^+,\kappa)\notcc(\aleph\_1,\aleph\_0)$ when $\kappa$ is supercompact. The…

Answering a question of Ketonen from the late 1970's, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing the limit of a decreasing sequence of…

We show that for many pairs of infinite cardinals $\kappa > \mu^+ > \mu$, $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent,…

We show that in the $\mathbb{P}_{\max}$ extension of a certain Chang-type model of determinacy, if $\kappa\in\{\omega_1, \omega_2, \omega_3\}$, then the restriction of the club filter on $\kappa\cap\mathrm{Cof}(\omega)$ to HOD is an…

Usuba has asked whether the $\kappa$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}\kappa$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $\kappa$ is a…

Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $\kappa$, there are no $\Sigma^1_1(\kappa)-\kappa-$mad families.

Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda > \kappa^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa) = \lambda$ starting from a ground model in…

Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_\kappa$ consisting of perfect trees of height $\lambda$ and width $\kappa$ where $\kappa$ is a singular $\omega$-strong limit of cofinality $\lambda$. They showed that if $\kappa$…

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly…

In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{\omega+1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable cardinal $\theta<\aleph_\omega$. In this…

Given an uncountable regular cardinal $\kappa$, a partial order is $\kappa$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $\kappa$ is stationary in $\mathcal{P}_\kappa(\mathbb{P})$. We…

We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal ${\kappa}\in On$, (1) ${\kappa}^+\to ({\kappa},{\omega}+1)$, (2) any…

Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is…