Big Ramsey degrees and structures in relational languages

We establish a precompact transfer theorem for big Ramsey structures and a uniform envelope theorem for valuation hypergraphs. These yield big Ramsey structures for generic injective relational structures in languages with finitely many symbols of each positive arity, answering the existence part of Problem 6.3.2 of Balko et al. (arXiv:2110.08409), and refute Conjecture 4.7 of Braunfeld et al. (arXiv:2301.13116). We also construct a finitely constrained free amalgamation class in a finite language of maximum arity three whose Fraïssé limit has infinite singleton big Ramsey degree, giving a negative answer to the existence question in Problem 6.3.3 of Balko et al.