Related papers: The (\lambda, \kappa)-Freese-Nation property for b…

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 >=…

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 $\mathbb{B}$…

Let $f:X\rightarrow Z$ be a separable fibration of relative dimension 1 between smooth projective varieties over an algebraically closed field $k$ of positive characteristic. We prove the subadditivity of Kodaira dimension…

We introduce the notion of L^2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that…

We show that, assuming GCH, if $\kappa$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and obeying the constraints of Easton's theorem, namely,…

Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable…

In this paper, we study the generalized (co)homology Hopf algebras of the loop spaces on the infinite classical groups, generalizing the work due to Kono-Kozima and Clarke. We shall give a description of these Hopf algebras in terms of…

Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $\kappa$ is a singular strong limit of uncountable cofinality, all scales on $\kappa$ are…

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to $\omega_1$-sequences of the selection principle and…

We discuss some notions of compactness and convergence relative to a specified family F of subsets of some topological space X. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in…

Let $\Gamma^\infty$ be the set of all universally Baire sets of reals. Inspired by recent work of the second author and Nam Trang, we introduce a new technique for establishing generic absoluteness results for models containing…

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…

It is shown that the fundamental group of the Griffiths double cone space is isomorphic to that of the triple cone. More generally if $\kappa$ is a cardinal such that $2 \leq \kappa \leq 2^{\aleph_0}$ then the $\kappa$-fold cone has the…

Let $\mu < \kappa < \lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u (\kappa, \lambda)$ is regular, where $u (\kappa, \lambda)$ denotes the least size of a…

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows…

In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where…

A Tychonoff space $X$ is called $\kappa$-pseudocompact if for every continuous mapping $f$ of $X$ into $\mathbb{R}^\kappa$ the image $f(X)$ is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying…

We prove that any suitable generalization of Laver forcing to the space $ \kappa^\kappa$, for uncountable regular $\kappa$, necessarily adds a Cohen $\kappa$-real. We also study a dichotomy and an ideal naturally related to generalized…

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…

In this paper we study the Borel reducibility of Borel equivalence relations, including some orbit equivalence relations, on the generalised Baire space $\kappa^\kappa$ for an uncountable $\kappa$ with the property…