Ramsey expansions for finitely homogeneous structures

Determine whether every structure that is homogeneous in a finite relational language is a reduct of an ω-categorical Ramsey structure; equivalently, ascertain whether the class of ω-categorical Ramsey structures, together with their reducts, includes all finitely homogeneous structures.

Background

The paper focuses on permutation modules over automorphism groups of ω-categorical structures, with particular emphasis on the case where the underlying structure is Ramsey (so its automorphism group is extremely amenable). The authors motivate this focus by noting that Ramsey structures form a broad and robust class supporting strong structural and algorithmic results.

They explicitly mention a community conjecture that the class under consideration should encompass all finitely homogeneous structures. Establishing this would significantly broaden the reach of their methods, since it would imply that permutation-module techniques developed here apply to every finitely homogeneous structure via suitable Ramsey expansions or reducts.