On the complexity of the relations of isomorphism and bi-embeddability

Abstract

Given an L ω 1 ω \mathcal {L}_{\omega _1\omega } -elementary class C \mathcal {C} , that is, the collection of the countable models of some L ω 1 ω \mathcal {L}_{\omega _1 \omega } -sentence, denote by ≅ C \cong _{\mathcal {C}} and ≡ C \equiv _{\mathcal {C}} the analytic equivalence relations of, respectively, isomorphisms and bi-embeddability on C \mathcal {C} . Generalizing some questions of A. Louveau and C. Rosendal, in a paper by S. Friedman and L. Motto Ros they proposed the problem of determining which pairs of analytic equivalence relations ( E , F ) (E,F) can be realized (up to Borel bireducibility) as pairs of the form ( ≅ C , ≡ C ) (\cong _{\mathcal {C}}, \equiv _{\mathcal {C}}) , C \mathcal {C} some L ω 1 ω \mathcal {L}_{\omega _1\omega } -elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such a problem: under very mild conditions on E E and F F , it is always possible to find such an L ω 1 ω \mathcal {L}_{\omega _1\omega } -elementary class C \mathcal {C} .