Related papers: On the cofinality of the least $\lambda$-strongly …
Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not…
We continue our study of the class $\mathscr{C}(D)$, where $D$ is a uniform ultrafilter on a cardinal $\kappa$ and $\mathscr{C}(D)$ is the class of all pairs $(\theta_1, \theta_2),$ where $(\theta_1, \theta_2)$ is the cofinality of a cut in…
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular…
An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa.$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where…
We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold…
Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg…
A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative…
Given two infinite cardinals $\kappa$ and $\lambda$, we introduce and study the notion of a $\kappa$-barely independent family over $\lambda.$ We provide some conditions under which these types of families exist. In particular, we relate…
We give some general criteria, when kappa-complete forcing preserves largeness properties -- like kappa-presaturation of normal ideals on lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the…
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 $\kappa$…
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…
This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$,…
This short note contains the proofs of two small but somewhat surprising results about ultrafilters on $\mathbb{N}$: 1. strongly summable ultrafilters are rapid, 2. every rapid ultrafilter induces a closed left ideal of rapid ultrafilters.…
We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a…
We show that if $\lambda^{<\kappa} = \lambda$ and every normal filter on $P_\kappa\lambda$ can be extended to a $\kappa$-complete ultrafilter then so does every $\kappa$-complete filter on $\lambda$. This answers a question of Gitik.
We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving…
If kappa is strongly compact, lambda > kappa is regular, then (2^{< lambda})^+ --> (lambda+eta)^2_theta holds for eta,theta<kappa.
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.
Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying…
Building on work of Holy, L\"ucke and Njegomir \cite{MR3913154} on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see \cite{MR0384553} and \cite{MR0540770}), to give characterizations of…