Related papers: Characterizations of the weakly compact ideal on $…

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order…

Given an ideal $I$ on $\omega$ let $a(I) $ ($\bar{a}(I)$) be minimum of the cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of $[{\omega}]^{\omega}$, and denote $b_I$ and$d_I$ the unbounding and dominating…

For a topological space $X$ its reflection in a class $\mathsf T$ of topological spaces is a pair $(\mathsf T X,i_X)$ consisting of a space $\mathsf T X\in\mathsf T$ and continuous map $i_X:X\to \mathsf T X$ such that for any continuous map…

We prove that the existence of a non-special tree of size $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no $K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the special tree number…

We prove that for regular $\lambda$ above a strong limit singular $\mu$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $\lambda$ and $\mu$ there are coboundedly many regular…

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…

Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$…

We introduce a covering notion depending on two cardinals, which we call $\mathcal O $-$ [ \mu, \lambda ]$-compactness, and which encompasses both pseudocompactness and many other generalizations of pseudocompactness. For Tychonoff spaces,…

We prove that for infinite cardinals $\kappa<\lambda$ the alternating group $Alt(\lambda)$ (of even permutations) of $\lambda$ is not embeddable into the symmetric group $Sym(\kappa)$ (of all permutations) of $\kappa$. To prove this fact we…

We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the…

Given a cardinal $\kappa$ and filters $\mathcal{F}_\alpha$ on $\omega$ for $\alpha\in\kappa$, we will show that if $\prod_{\alpha\in\kappa}\mathcal{F}_\alpha$ is countable dense homogeneous then $\kappa<\mathfrak{p}$ and each…

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible…

In 1994, Jan Pelant proved that a metric property related to the notion of paracompactness called the uniform Stone property characterizes a metric space's uniform embeddability into $c_0(\kappa)$ for some cardinality $\kappa$. In this…

Let $\mathrm{cof}(\mu)=\mu$ and $\kappa$ be a supercompact cardinal with $\mu<\kappa$. Assume that there is an increasing and continuous sequence of cardinals $\langle\kappa_\xi\mid \xi<\mu\rangle$ with $\kappa_0:=\kappa$ and such that, for…

We discuss some well-known compactness principles for uncountable structures of small regular sizes ($\omega_n$ for $2 \le n<\omega$, $\aleph_{\omega+1}$, $\aleph_{\omega^2+1}$, etc.), consistent from weakly compact (the size-restricted…

For $\kappa$ regular and uncountable we define variants of the classical cardinal characteristics modulo the non-stationary ideal.

A ballean is a set $X$ endowed with some family $\F$ of its subsets, called the balls, in such a way that $(X,\F)$ can be considered as an asymptotic counterpart of a uniform topological space. Given a cardinal $\kappa$, we define $\F$…

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally…

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $\kappa$ is indestructible by the higher random forcing $\mathbb Q_\kappa$. We then use this characterisation to show that…

We prove that regular supercuspidal representations of $p$-adic groups are uniquely determined by their character values on very regular elements -- a special class of regular semisimple elements on which character formulae are very simple…