The classifiability and wandering-types characterization of the Schröder–Bernstein property

Throughout, $T$ is a complete first-order theory. Say that $T$ has the Schröder–Bernstein property if elementarily bi-embeddable models of $T$ are isomorphic. A type $p\in S(M)$ is wandering when there are $M\models T$ and $f\in\operatorname{Aut}(M)$ such that

$$f^i(p)\mathrel{\bot^a}f^j(p)$$

for every $i<j<\omega$.

The classifiability and wandering-types conjecture. The complete theory $T$ has the Schröder–Bernstein property if and only if it is classifiable and has no wandering types.

This is presented as the paper's main proposed characterization. The authors note that the necessary classifiability condition is known at least for countable theories, and that theories with the Schröder–Bernstein property cannot have nomadic types; the full characterization remains open, with special cases known for certain locally modular groups and weakly minimal theories of abelian groups.