Big Ramsey structure for the generic 3-uniform hypergraph

Characterize the big Ramsey degrees of the generic 3-uniform hypergraph and determine whether it admits a big Ramsey structure.

Background

The paper notes that no structure with a relation of arity greater than two was then known to admit a big Ramsey structure. The generic 3-uniform hypergraph is presented as the first basic higher-arity test case.

References

Characterize the big Ramsey degrees of the generic $3$-uniform hypergraph. Does the generic $3$-uniform hypergraph 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