Related papers: Superclub, splitting, separating statements

We investigate the unbalanced ordinary partition relations of the form $\lambda \rightarrow {(\lambda, \alpha)}^{2}$ for various values of the cardinal $\lambda$ and the ordinal $\alpha$. For example, we show that for every infinite…

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa,\lambda$ are infinite cardinals such that $\kappa^{+++} \leq \lambda$, $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}= \kappa^+$, and…

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $\lambda$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $\kappa$ is a tall…

This note shows how recent work of Eskew and Hayut can be combined with a method of Raghavan and Shelah to show the consistency of $\mathfrak u_\kappa<2^\kappa$ for $\aleph_3\leq \kappa<\aleph_{\omega}$, from a huge cardinal.

Let $G$ be an infinite compact group. We prove that for every cardinal $\kappa$ between the density and the weight of $G$, there exists a dense subgroup of $G$ of density $\kappa$.

Assuming that $GCH$ holds and $\kappa$ is $\kappa^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $\kappa$ remains strongly inaccessible and $(\alpha^+)^{HOD} < \alpha^+$ for every infinite cardinal $\alpha <…

For any regular cardinal $\kappa$ and ordinal $\eta<\kappa^{++}$ it is consistent that $2^{\kappa}$ is as large as you wish, and every function $f:\eta \to [\kappa,2^{\kappa}]\cap Card$ with $f(\alpha)=\kappa$ for $cf(\alpha)<\kappa$ is the…

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…

After small forcing, any < kappa-closed forcing will destroy the supercompactness, even the strong compactness, of kappa .

We prove the following continuous analogue of Vaught's Two-Cardinal Theorem: if for some $\kappa>\lambda\geq \aleph_0$, a continuous theory $T$ has a model with density character $\kappa$ which has a definable subset of density character…

In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say theta strongly non-reflects at lambda iff there is a function F: theta ---> lambda such that for all alpha < theta with cf(alpha)=…

Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal $\kappa$ becomes superdestructible---any further ${<}\kappa$-closed forcing which adds a subset to…

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. We answer the longstanding question, whether $\kappa = \kappa^{<\kappa} \geq\aleph_1$…

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:…

Motivated by the goal of constructing a model in which there are no $\kappa$-Aronszajn trees for any regular $\kappa>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square…

Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.

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 deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S…

We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $ \lambda $ is a singular cardinal, $ \lambda ' < \lambda $, and the ultrafilter $D$ is $ \kappa $-decomposable for all…

We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…