The tractability conjecture for reducts of finitely bounded homogeneous structures

Let ${\mathbb A}$ be a reduct of a finitely bounded homogeneous structure: it is obtained by forgetting some relations from a homogeneous structure, and the ambient structure is characterized among finite structures by forbidding embeddings of a finite set of finite structures. Let ${\mathbb K}_3$ denote the complete graph on three vertices, and let $\operatorname{CSP}({\mathbb A})$ be the problem of deciding whether a finite input structure admits a homomorphism to ${\mathbb A}$.

The tractability conjecture. If ${\mathbb A}$ does not pp-construct ${\mathbb K}_3$, then $\operatorname{CSP}({\mathbb A})$ is polynomial-time solvable.

This is described as a wide-open generalization of the Feder–Vardi conjecture to reducts of finitely bounded homogeneous structures and is the subject of an active research program.