# 2010 seminar talk: On the complexity of the relations of isomorphism and bi-embeddability

Talk held by Luca Motto Ros (KGRC) at the KGRC seminar on 2010-01-28.

### Abstract

Given a pair of analytic equivalence relations E and F, we determine under
which conditions it is possible to find an L_{ω1,ω}-elementary class
C such that E and F are Borel equivalent to, respectively, isomorphism and bi-
embeddability on C: quite surprisingly, it turns out that, apart from the
obvious limitations, such an elementary class always exists. This result gives
an (almost complete) answer to previous questions of Louveau-Rosendal (2005)
and Friedman-Motto Ros (2009).