Polynomial-time homomorphism-embedding CSP representation

Determine whether every homogeneous Ramsey structure in a finite relational language is first-order bi-interpretable with a homogeneous finite-relational-language Ramsey structure whose homomorphism-embedding constraint satisfaction problem is solvable in polynomial time.

Background

The conjecture is motivated by the observation that completion methods often yield polynomial-time algorithms for homomorphism-embedding CSPs. The paper also gives a model-complete-core refinement.

References

Every homogeneous Ramsey structure $\str M$ in a finite relational language is first-order bi-interpretable with a homogeneous Ramsey structure $\str H$ in a finite relational language such that the homomorphism-embedding variant of the constraint satisfaction problem of $\str H$ is solvable in polynomial time.

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Conjecture, subsection Ramsey classes