Related papers: On inverse gamma-systems and the number of L_{inft…

In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square(\lambda^+)$. Furthermore, we establish…

It is well-known that a finitely generated group $\Gamma$ has Kazhdan's property (T) if and only if the Laplacian element $\Delta$ in ${\mathbb R}[\Gamma]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in…

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n, p\geq2$, $\Gamma$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is…

The main result is Theorem: Let A be an R-algebra, mu, lambda be cardinals such that |A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is countably free, respectively, then there exists an aleph_0-cotorsion-free or a…

Let $\kappa$ be an uncountable cardinal with $\kappa=\kappa^{{<}\kappa}$. Given a cardinal $\mu$, we equip the set ${}^\kappa\mu$ consisting of all functions from $\kappa$ to $\mu$ with the topology whose basic open sets consist of all…

Let $\kappa$ be a regular cardinal. Consider the Baire numbers of the spaces $(2^{\theta})_\kappa$ (functions from $\theta$ to 2 and the less than $\kappa$ topology) for various $\theta \geq \kappa$. Let l be the number of such different…

A permutation group $G$ on a set $A$ is ${\kappa}$-homogeneous iff for all $X,Y\in [A]^{\kappa}$ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. $G$ is ${\kappa}$-transitive iff for any injective function $f$…

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

We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\…

We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the…

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.

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…

We investigate in ZFC what can be the family of large enough cardinals mu in which an a.e.c. K is categorical or even just solvable. We show that for not few cardinals lambda<mu there is a superlimit model in K_lambda. Moreover, our main…

Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical…

For $\mu, \kappa$ infinite, say $\mathcal{A}\subseteq [\kappa]^\kappa$ is a $(\mu,\kappa)$-maximal independent family if whenever $\mathcal{A}_0$ and $\mathcal{A}_1$ are pairwise disjoint non-empty in $[\mathcal{A}]^{<\mu}$ then…

If $V = L$, and $\mu$, $\kappa$ and $\lambda$ are three infinite cardinals with $\mu = {\rm cf} (\mu) < \kappa = {\rm cf}(\kappa) \leq \lambda$, then, as shown in \cite{Heaven}, the $\mu$-club filters on $P_\kappa (\lambda)$ and $P_\kappa…

We show that if $\gamma \in \omega \cup \{\aleph_{0}\}$ and $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture having exactly $\gamma$ many models of Scott rank $\omega_{1}$, then there exists a club $C \subseteq…

We prove that every point-finite family of nonempty functionally open sets in a topological space $X$ has the cardinality at most an infinite cardinal $\kappa$ if and only if $w(X)\leq\kappa$ for every Valdivia compact space $Y\subseteq…

We point out a gap in Shelah's proof of the following result: $\mathbf{Claim}$ Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal $\lambda$ such that whenever $M, N \in K$ have…

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…