Related papers: Universally Baire sets in $2^{\kappa}$

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…

It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0}…

It is known that the large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is much stronger than the Axiom of Determinacy itself. Sargsyan conjectured it to be as…

We construct a model in which the continuum has size $\kappa$ for a regular cardinal $\kappa$ and in which the $\Sigma^1_n$-uniformization property holds simultaneously for every $n \ge 2$. Additionally this model has a $\Delta^1_3$-…

Suppose $\kappa$ is a regular cardinal and $\bar a=\langle \mu_i: i<\kappa \rangle$ is a non-decreasing sequence of regular cardinals. We study the set of possible cofinalities of cuts Pcut$(\bar a)=\{(\lambda_1, \lambda_2):$ for some…

We consider two quantities that measure complexity of binary strings: $\mathit{KA}(x)$ is defined as the minus logarithm of continuous a priori probability on the binary tree, and $\mathit{KP}(x)$ denotes prefix complexity of a binary…

A $r$-parameter ${u}_{\{\kappa_1, \kappa_2, \cdots, \kappa_r\}}(2)$ algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl-Heisenberg…

For an arbitrary state $\omega$ on a Cuntz algebra, we define a number $1\leq \kappa(\omega)\leq \infty$ such that if the GNS representations of $\omega$ and $\omega'$ are unitarily equivalent, then $\kappa(\omega)=\kappa(\omega')$. By…

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For $n\ge 2$, there is a…

The additivity spectrum ADD(I) of an ideal I is the set of all regular cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\} in the ideal I such that the union of the chain is not in I. We investigate which set A of…

We present some results about the burgeoning research area concerning set theory of the kappa-reals. We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly…

We investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7].…

Let $\mathbb{B}$ be a complete Boolean algebra. We show, as an application of a previous result of the author, that if $\lambda$ is an infinite cardinal and $\mathbb{B}$ is weakly $(\lambda^\omega, \omega)$-distributive, then $\mathbb{B}$…

A subset $R$ of the vertex set of a graph $\Gamma$ is said to be $(\kappa,\tau)$-regular if $R$ induces a $\kappa$-regular subgraph and every vertex outside $R$ is adjacent to exactly $\tau$ vertices in $R$. In particular, if $R$ is a…

A set $U\subseteq \reals^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \subseteq \reals^2$ consists of points chosen randomly and uniformly from the unit square then…

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show that the elementary presentation of the Van Kampen condition is actually a characterisation of a universal property in the associated bicategory…

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…

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…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…

Given a weakly compact cardinal $\kappa$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{\kappa^+, \kappa}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the…