J. Symbolic Logic 76 (1), 243-266, (March 2011) DOI: 10.2178/jsl/1294170999
Sy-David Friedman, Luca Motto Ros
KEYWORDS: Analytic equivalence relation, analytic quasi-order, bi-embeddability, Borel reducibility, 03E15
Louveau and Rosendal  have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ℒω1ω) is far from complete (see [5, 2]).
In this article we strengthen the results of  by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in  about the comparison between isomorphism and bi-embeddability. Finally, as in  our results apply not only to classes of countable structures defined by sentences of ℒω1ω, but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.