Big Ramsey analogue of the Nešetřil–Rödl theorem

Characterize the finite collections of finite irreducible structures in a finite relational language whose free-class Fraïssé limit has finite big Ramsey degrees or admits a big Ramsey structure.

Background

This is proposed as a general infinitary analogue of the Nešetřil–Rödl theorem for free amalgamation classes. It is intended to follow the resolution of several concrete higher-arity examples.

References

Let $L$ be a finite relational language. Characterize those finite collections of finite irreducible $L$-structures $\mathcal F$ for which the limit of the class of all finite $\mathcal F$-free structures has finite big Ramsey degrees, resp. admits a big Ramsey structure.

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