Sparse ordered 3-uniform hypergraphs with interval chromatic number

Determine whether, for fixed positive integers $d$ and $\chi$, there exists $\varepsilon=\varepsilon(d,\chi)>0$ such that every ordered 3-uniform hypergraph $H^<$ on $n$ vertices with maximum degree $d$ and interval chromatic number $\chi$ satisfies $R_<(H^<)\le2^{O(n^{2-\varepsilon})}$.

Background

A subquadratic-exponent upper bound is known when the interval chromatic number is three. The problem asks whether a comparable bound holds for every fixed interval chromatic number while retaining bounded maximum degree.

References

Another interesting problem is to extend the upper bound with subquadratic exponent from Theorem~\ref{thm-ordRamHyper-3UnifMaxDegIntChr} to ordered $3$-uniform hypergraphs with bounded maximum degree and fixed interval chromatic number that is larger than $3$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem in Subsection “Bounded degrees and interval chromatic number” under Section “Hypergraph Ordered Ramsey Numbers”