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

Given an L_{ω_1 ω}-elementary class C, that is the collection of the countable models of some L_{ω_1 ω}-sentence, denote by \cong_C and \equiv_C the analytic equivalence relations of, respectively, isomorphism and bi-embeddability on C. Generalizing some questions of Louveau and Rosendal [LR05], in [FMR09] it was proposed the problem of determining which pairs of analytic equivalence relations (E,F) can be realized (up to Borel bireducibility) as pairs of the form (\cong_C,\equiv_C), C some L_{ω_1 ω}-elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such problem: under very mild conditions on E and F, it is always possible to find such an L_{ω_1 ω}-elementary class C.

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