In this article we survey the emerging field of descriptive graph combinatorics. This area has developed in the last two decades or so at the interface of descriptive set theory and graph theory, and it has interesting connections with other areas such as ergodic theory and probability theory. Our object of study is the theory of definable graphs, usually Borel or analytic graphs on Polish spac...

We construct two Borel equivalence relations on the generalized Baire space κ, κ = κ > ω, with the property that neither of them is Borel reducible to the other. A small modification of the construction shows that the straightforward generalization of the Glimm-Effros dichotomy fails. By λ we denote the set of all functions κ→ λ. We define a topology to (λ) by letting the sets N(η1...,ηn) = {(f...

This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a B...

2 Euler characteristic analysis 14 2.1 Formulation of the result and the nature of the generating function . . . . . 14 2.2 Standard Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Arbitrary Borel in gl(N |N). . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Arbitrary Borel in gl(M |N). . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Arb...

We introduce the notion of Q-Borel ideals: ideals which are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals.