Big Ramsey degrees for a mixed binary-ternary forbidden structure

Determine whether the Fraïssé limit of finite structures omitting the specified four-vertex mixed binary-ternary structure has finite big Ramsey degrees and a big Ramsey structure.

Background

The problem concerns a finite relational language with one binary and one ternary relation and forbids embeddings of a specific four-vertex structure. It is another test case for extending big Ramsey methods beyond binary free amalgamation.

References

Let $L={E,H}$ be a language with one binary relation $E$ and one ternary relation $H$. Let $\str{F}$ be the $L$-structure where $F={0,1,2,3}, E_\str F={(1,0),(1,2),(1,3)}, H_\str F={(0,2,3)}$. Denote by $\mathcal K$ the class of all $L$-structures $\str{A}$ such that there is no monomorphism $\str{F}\to\str{A}$. Does the limit of $\mathcal K$ have finite big Ramsey degrees and a big Ramsey structure?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Problem 1.4, subsection Big Ramsey structures