Isomorphism of Homogeneous Structures

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 for transitive countable structures is as complicated as it is for arbitrary countable structures. We then use these results to characterize the complexity of the isometry relation for certain classes of homogeneous and ultrahomogeneous metric spaces. In their article [4], Friedman and Stanley considered the question of how difficult it is to classify a collection of countable first-order structures up to isomorphism. To make this precise, they define the space of countable models of a given first-order theory, and consider the isomorphism relation as an equivalence relation on this space. They then use the relation of Borel reducibility of equivalence relations to compare such isomorphism relations, thus characterizing the difficulty of the corresponding isomorphism problem. Certain first-order languages and theories have an isomorphism problem of maximal complexity, in the sense that any other such isomorphism relation can be reduced to them. Such theories are called Borel-complete. Many of the techniques for showing that a given theory is Borel-complete involve coding other structures into models of the given theory, and this generally involves the use of distinguished points or definable subsets in the models produced. The aim of this article is to consider the extent to which distinguished points can be eliminated, that is, to consider the complexity of the isomorphism problem for structures with no distinguished points. Printed November 11, 2008 2001 Mathematics Subject Classification: Primary, 03E15; Secondary, 03C15, 03C50