Big Ramsey degrees for finite relational languages of arbitrary arity

Characterize the big Ramsey degrees of the Fraïssé limit of all finite structures in a relational language having finitely many relation symbols of each arity and injective relation tuples, and determine whether the limit admits a big Ramsey structure.

Background

This generalizes the generic 3-uniform hypergraph problem to arbitrary finite relational signatures with finitely many symbols at each arity.

References

Given a relational language $L$ with finitely many relational symbols of each arity, characterize the big Ramsey degrees for the class $\mathcal K$ of all finite $L$-structures where tuples in each relations are injective. Does the limit of $\mathcal K$ admit a big Ramsey structure?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Problem 7.2, subsection Big Ramsey structures