Related papers: Small u(kappa) at singular kappa with compactness …

We show that there are locally compact spaces that can be condensed onto separable spaces but not onto compact separable spaces. We also show that for every cardinal $\kappa$ there is a locally compact topological group of cardinality…

Given a strong limit cardinal $\lambda$ of countable cofinality, we show that if every $\lambda$-coanalytic subset of the generalised Cantor space ${}^{\lambda}2$ has the $\lambda$-$\mathsf{PSP}$, then there is an inner model with…

We study conditions on automorphisms of Boolean algebras of the form $P(\lambda)/I_\kappa$ (where $\lambda$ is an uncountable cardinal and $I_\kappa$ is the ideal of sets of cardinality less than $\kappa$) which allow one to conclude that a…

We prove that the existence of a complete metric space of cardinality at most $2^{\kappa}$ admitting Kuratowski partition is a consequence of $\kappa$ being the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$.

We analyze the intermediate models of the strongly compact Prikry forcing. We exhibit a simple combinatorial property which, for a given supercompact cardinal $\kappa$, characterize the projections of all projections of the strongly compact…

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$. In addition, we prove several global results…

We characterize the compactness properties of the product of \lambda\ copies of the space \omega\ with the discrete topology, dealing in particular with the case \lambda\ singular, using regular and uniform ultrafilters, infinitary…

Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality \kappa. We denote by \mu (respectively \hat\mu) the number of…

We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >=…

We show that compact cardinals and {\rm MM} are sensitive to $\lambda$-closed forcings for arbitrarily large $\lambda$. This is done by adding 'regressive' $\lambda$-Kurepa-trees in either case. We argue that the destruction of regressive…

The $\kappa$-density of a cardinal $\mu\ge\kappa$ is the least cardinality of a dense collection of $\kappa$-subsets of $\mu$ and is denoted by $\mathcal D(\mu,\kappa)$. The Singular Density Hypothesis (SDH) for a singular cardinal $\mu$ of…

For any cardinal $\kappa \geq 2$, there is a unique complete real tree whose points all have valence $\kappa$. In this note, we show that, when $\kappa \geq 3$, it is necessary to assume completeness. More precisely, we show that there…

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…

Saturation is (mu,kappa)-transferable in T if and only if there is an expansion T_1 of T with |T_1| = |T| such that if M is a mu-saturated model of T_1 and |M| \geq kappa then the reduct M|L(T) is kappa-saturated. We characterize theories…

Let $\kappa$ be an infinite cardinal. A topological space $X$ is $\kappa$-bounded if the closure of any subset of cardinality $\le\kappa$ in $X$ is compact. We discuss the problem of embeddability of topological spaces into Hausdorff…

In this paper, we prove that if $\kappa$ is a almost strongly compact cardinal, then any MAEC with L\"owenheim-Skolem number below $\kappa$ is $<\kappa$-d-tame.

It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklossy and the first author, we show that this statement fails for…

We show that for any regular cardinal $\kappa$, $\square_{\kappa, 2}$ is consistent with "all $\kappa^+$-Aronszajn trees are special." By a result of Shelah and Stanley this is optimal in the sense that $\square_{\kappa, 2}$ may not be…

We show that if $\mu \leq \cf \lambda $ and $\lambda$ is a strong limit singular cardinal, then $[\mu, \lambda ]$-compactness is productive if and only if either $\mu= \omega $, or $\mu$ is $\lambda$-strongly compact.

We show that the reduced cofinality of the nonstationary ideal NS_kappa on a regular uncountable cardinal kappa may be less than its cofinality, where the reduced cofinality of NS_kappa is the least cardinality of any family F of…