Recent developments in the theory of Borel reducibility (Q2773236)

In the paper the authors present the proofs of the results announced by the authors in ``New dichotomies for Borel equivalence relations'' [Bull. Symb. Log. 3, 329-346 (1997; Zbl 0889.03038)]. Namely they prove two dichotomy theorems (labeled the sixth and the seventh theorems in the cited paper). The first of them states that if \(E\) is a Borel equivalence relation which is Borel reducible to the relation \(E_3\), the product of countably many copies of the Vitali equivalence relation \(E_0\), then \(E\) is Borel reducible to \(E_0\) or \(E\) is equivalent to \(E_3\) with respect to Borel reducibility. The second theorem states that if \(E\) is a Borel equivalence relation Borel reducible to the orbit equivalence relation \(E^X_G\) of a closed subgroup \(G\) of the infinite symmetric group for a Borel \(G\)-space \(X\) and \(E^X_G\) is Borel then \(E\) is Borel reducible to a countable Borel equivalence relation or \(E_3\) is Borel reducible to \(E\).