Related papers: The nonstationary ideal on P_\kappa(lambda) for la…

Generalizing Keisler's notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on $\omega_1$ must be somewhere $\omega_1$-dense. We prove a dichotomy…

We prove that some natural "outside" property is equivalent (for a first order class) to being stable. For a model, being resplendent is a strengthening of being kappa-saturated. Restricting ourselves to the case kappa > |T| for…

Given an arbitrary measurable cardinal $\kappa$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $\kappa$ is constructed.

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 show that it is consistent relative to a huge cardinal that for all infinite cardinals $\kappa$, $\square_\kappa$ holds and there is a stationary $S \subseteq \kappa^+$ such that $\mathrm{NS}_{\kappa^+} \restriction S$ is…

We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $\kappa$ is a limit of measurables then there is no wellorder of a subset of $P(\kappa)$ of length…

We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam's theorem and its extension by Hajnal,…

The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M…

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…

We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection…

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…

We investigate the extent to which ultrapowers by normal measures on $\kappa$ can be correct about powersets $\mathcal{P}(\lambda)$ for $\lambda>\kappa$. We consider two versions of this questions, the capturing property…

We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal $\kappa$ such that $2^\kappa > kappa^+$ then there…

We investigate whether the ultrafilter number function $\kappa \mapsto \mathfrak{u}(\kappa)$ on the cardinals is monotone, that is, whether $\mathfrak{u}(\lambda) \le \mathfrak{u}(\kappa)$ holds for all cardinals $\lambda < \kappa$ or not.…

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…

We introduce the notion of $K$-ideals associated with Kuratowski partitions and we prove that each $\kappa$-complete ideal on a measurable cardinal $\kappa$ can be represented as a $K$-ideal. Moreover, we show some results concerning…

If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing…

For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^\kappa\kappa$ and ${}^\kappa2$ (endowed with the ${<}\kappa$-box topology) have been relatively well-studied, but less is known about the case where…

Assuming the existence of a strong cardinal $\kappa$, a weakly compact cardinal $\lambda$ above it and $\gamma > \lambda,$ we force a generic extension in which $\kappa$ is a singular strong limit cardinal of any given cofinality $\delta$,…

Let $\mathcal M_X$ denote the ideal of meager subsets of a topological space $X$. We prove that if $X$ is a completely metrizable space without isolated points, then the smallest cardinality of a non-meager subset of $X$, denoted…