We prove that for every Borel equivalence relation E, either E is Borel reducible to E0, or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F is a Borel equivalence relation and F is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence r...

Quasi-hereditary algebras arise in several natural contexts. For example, their module categories often appear in connection with the representation theory of algebraic groups over elds of positive characteristic or of quantum groups at a root of unity. This point of view has been exploited by various authors, e. g., [CPS3{8], [G2], etc. Classical Schur algebras attached to the general linear g...

Given a countable Borel equivalence relation E on a Polish space, let IE denote the σ-ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism from IE to IF if and only if there is a smooth-to-one Borel homomorphism from a finite index Borel subequivalence relation of E to F . As a corollary, we see that IE is homogeneous in the sense of Zapletal (2007,...

Families of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering, ≤B , are constructed. There is an analytic ideal on ù generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spa...

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal in [8] for the study of analytic equivalence relations, we study countable Bo...

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equ...

Throughout, by a graph we mean a simple undirected graph, where the degree of a vertex is its number of neighbors, and a d-coloring is a function assigning each vertex one of d colors so that adjacent vertices are mapped to different colors. This paper examines measurable analogues of Brooks’s Theorem. While a straightforward compactness argument extends Brooks’s Theorem to infinite graphs, suc...

Let E ⊆ F and E′ ⊆ F ′ be Borel equivalence relations on the standard Borel spaces X and Y , respectively. The pair (E,F ) is simultaneously Borel reducible to the pair (E′, F ′) if there is a Borel function f : X → Y that is both a reduction from E to E′ and a reduction from F to F ′. Simultaneous Borel embeddings and isomorphisms are defined analogously. We classify all pairs E ⊆ F of smooth ...