The homogeneous-structure characterization of measurable ultraproduct classes

Let $L$ be a finite relational language. A countable $L$-structure is homogeneous if every isomorphism between finite substructures extends to an automorphism. A class is an m.e.c. if it is a multidimensional exact class, and an ultraproduct is formed from members of that class.

Homogeneous-structure conjecture. (i) Let $M$ be a homogeneous structure over a finite relational language $L$. Then there is an m.e.c. with ultraproduct elementarily equivalent to $M$ if and only if $M$ is stable.

(ii) Let $\mathcal{C}$ be an m.e.c. and let $M$ be an unstable homogeneous structure over a finite relational language. Then $M$ is not elementarily equivalent to any structure interpretable in an ultraproduct of $\mathcal{C}$.

This conjecture characterizes which homogeneous structures can arise, up to elementary equivalence, from ultraproducts of multidimensional exact classes. The paper notes that a general proof may require revisiting Lachlan's work on finite homogeneous structures under weaker assumptions than full homogeneity; the conjecture also motivates questions about highly regular finite structures without strong automorphism-group hypotheses.