Related papers: A Small Ultrafilter Number at Every Singular Cardi…

Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…

For an uncountable regular cardinal \kappa we let \nabla_\kappa(A) be the statement that A \subset \kappa and for all regular \theta > \kappa, the set of all X \in [\theta]^<\kappa such that X \cap \kappa \in \kappa and otp(X \cap OR) is a…

This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such…

We study the generalized dominating number $\mathfrak{d}_{\mu}$ at a singular cardinal $\mu$ of cofinality $\kappa$. We show two lower bounds: in ZFC, $\mathrm{cf}([\mu]^\kappa,\subseteq) \leq \mathfrak{d}_{\mu}$, and under mild…

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma_1$-definability at uncountable regular cardinals. In particular we give its exact consistency strength firstly in terms of the second…

We prove that \textsf{P}-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph_{\omega}% $-many random reals to a model of \textsf{CH. }These results improve and correct…

We investigate the circumstances under which there exist a singular cardinal $\mu$ and a short $(\kappa, \mu)$-extender $E$ witnessing "$\kappa$ is $\mu$-strong", such that $\mu$ is singular in $\Ult(V, E)$.

We show that the tree property, stationary reflection and the failure of approachability at $\kappa^{++}$ are consistent with $\mathfrak{u}(\kappa) = \kappa^+ < 2^\kappa$, where $\kappa$ is a singular strong limit cardinal with the…

We continue the study from \cite{BrendleFreidmanMontoya, vandervlugtlocalizationcardinals} of localization cardinals $\mfb_\kappa(\in^*)$ and $\mfd_\kappa(\in^*)$ and their variants at regular uncountable $\kappa$. We prove that if $\kappa$…

In this paper, we demonstrate that if, for every $\kappa$-complete fine filter $F$ over $\mathcal{P}_{\kappa}\lambda$, the associated Namba forcing $\mathrm{Nm}(\kappa,\lambda,F)$ is semiproper, then $\square(\mu,{<}\aleph_1)$ fails for all…

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong…

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally…

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are…

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows…

We study the relations between a generalization of pseudocompactness, named $(\kappa, M)$-pseudocompactness, the countably compactness of subspaces of $\beta \omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the…

For an ultrafilter $D$ on a cardinal $\kappa,$ we wonder for which pair $(\theta_1, \theta_2)$ of regular cardinals, we have: for any $(\theta_1+\theta_2)^+-$saturated dense linear order $J, J^{\kappa}/ D$ has a cut of cofinality…

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\lambda\leq \kappa$, the following theories are equiconsistent modulo ZFC:…

It is shown that the consistency strength of ZF + DC + "the closed unbounded ultrafilter on omega_1 is an ultrafilter" is exactly ZFC + one measurable cardinal.

We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{\kappa, \kappa}$, this allows us to characterize any large cardinal…

Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…