Related papers: Cardinal characteristics and countable Borel equiv…
In this paper, we study some properties of associated sequaences in umbral calculus. From these properties, we derive new and interesting identities of several kinds of polynomials.
There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles.…
We examine the relation of BSS-reducibility on subsets of the real numbers. The question was asked recently (and anonymously) whether it is possible for the halting problem H in BSS-computation to be BSS-reducible to a countable set.…
If the Continuum Hypothesis is false, it implies the existence of cardinalities between the integers and the real numbers. In studying these "cardinal characteristics of the continuum", it was discovered that many of the associated…
We provide a comprehensive development of the basics of descriptive set theory for non-separable complete metric spaces whose weight is a singular cardinal $\lambda$ of countable confinality. Somewhat unexpectedly, the resulting theory is…
We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is…
We give new equivalent characterizations for ideals of Borel type. Also, we prove that the regularity of a product of ideals of Borel type is bounded by the sum of the regularities of those ideals.
We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an…
The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A $\mathrm{ZFC}$ lower bound involving the minimum of the classical almost disjointness and…
We provide a pure algebraic version of the dynamical characterization of Conrad's property. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof…
We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable $\mathcal{L}_{\omega_1\omega}$ theories which admit a one-sorted interpretation of a particular theory we call…
This article is an introduction to combinatorics under the axiom of determinacy with a focus on partition properties and infinity Borel codes.
We solve four out of the six open problems concerning critical cardinalities of topological diagonalization properties involving tau-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this…
We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the 9 finite graphs from the classical result of Beineke together with a 10th infinite graph associated to the equivalence relation $\mathbb{E}_0$ on the…
We show that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\varepsilon > 0$…
We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of…
We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal…
We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_{\sigma}$…
The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal…
Recently, Chatterjee has introduced a new coefficient of correlation which has several natural properties. In particular, the coefficient attains its maximal value if and only if one variable is a measurable function of the other variable.…