Big Ramsey degrees of universal finite-distance metric spaces

Characterize the big Ramsey degrees of the countable universal metric space with a fixed finite distance set S and determine whether it admits a big Ramsey structure.

Background

Existing methods establish finiteness for homogeneous metric spaces with finitely many distances, but the bounds are far from optimal. Exact degrees and canonical expansions are sought to clarify the underlying combinatorics.

References

Let $S\subseteq \mathbb R{>}0}$ be finite and let $\mathbb U_S$ be a countable metric spaces with distances from $S$ universal for all countable metric spaces with distances from $S$. Characterise the big Ramsey degrees of $\mathbb U_S$. Does it 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, subsection Big Ramsey structures