We consider the complexity of the isomorphism relation on countable first-order structures with transitive automorphism groups. We use the theory of Borel reducibility of equivalence relations to show that the isomorphism problem for vertex-transitive graphs is as complicated as the isomorphism problem for arbitrary graphs, and determine for which first-order languages the isomorphism problem f...

We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of E1 into E. Our techniques also yield the first classical proof of the analogous result for hypers...

We prove that if X is a locally compact σ-compact space then on its quotient, γ(X) say, determined by the algebra of all real valued bounded continuous functions on X, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if X is second countable locally compact then γ(X) is second countable locally compact Hausdorff if and only i...

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite str...

We study the class of languages of finitely-labelled countable linear orderings definable in twovariable first-order logic. We give a number of characterisations, in particular an algebraic one in terms of circle monoids, using equations. This generalises the corresponding characterisation, namely variety DA, over finite words to the countable case. A corollary is that the membership in this cl...

Notes on set theory, mainly forcing. The first four sections closely follow the lecture notes Williams [8] and the book Kunen [4]. The last section covers topics from various sources, as indicated there. Hopefully, all errors are mine. 1. The Suslin problem 1.1. The Suslin hypothesis. Recall that R is the unique dense, complete and separable order without endpoints (up to isomorphism). It follo...

We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in אα for some (equivalently for every ordinal) α > 0. Another central result here is, in this context: the number of models of a countable first order T of cardinality אα is either ≥ |α| for every α or it has a small upper bound (independent of α close to i2). The autho...