Essentially countable isomorphism relations of first order theories

Given a theory $T$, $\operatorname{Mod}(T)$ denotes the Polish space of models of $T$ with domain $\mathbb{N}$ and $\cong_T$ denotes the isomorphism relation on $\operatorname{Mod}(T)$.

A Borel equivalence relation $E$ is essentially countable if it is Borel reducible to a countable Borel equivalence relation and smooth if it is Borel reducible to the equality relation on $2^\mathbb{N}$.

• It is well known that the question has a negative answer if countable $\mathcal{L}_{\omega_1, \omega}$-theories are allowed.
• Marker [Mar07] showed that if $T$ has uncountably many types then $\cong_T$ is not essentially countable and hence the question has a positive answer for such theories.
• Rast [Ras17] showed the question has a positive answer when $T$ is a colored linear order (i.e. a linear order with some unary predicates).
• Rast and Sahota [RS17] showed that the question has a positive answer for o-minimal theories.

The question was originally asked by Hjorth and Kechris in 2005 at the Notre Dame workshop on Vaught's Conjecture.