Ordered 3-uniform hypergraph gap

Close the gap between the lower and upper bounds on R_<(H^<,K^{<(3)}_3(n)) for an n-vertex ordered 3-uniform hypergraph H^< of fixed maximum degree d.

Background

For bounded-degree ordered 3-uniform hypergraphs, the survey gives superexponential lower bounds and subquadratic-exponent upper bounds for the off-diagonal problem against a tripartite ordered hypergraph. The problem asks to close the remaining exponent gap.

References

Theorems~\ref{thm-ordRamHyper-3UnifMaxDegIntChr} and~\ref{thm-ordRamHyper-3UnifMaxDegLower} give estimates on the ordered $R_<(G<, K{<(3)}_3(n))$ and although the exponents in the bounds are reasonably close, there is still a gap between them and it would be interesting to close it.

Let $d$ be a fixed positive integer and let $H<$ be an ordered $3$-uniform hypergraph on $n$ vertices with maximum degree $d$. Close the gap between the lower and upper bounds on $R_<(H<, K{<(3)}_3(n))$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Theorem (thm-ordRamHyper-3UnifMaxDegLower), Section 3.3

Close the gap between the lower and upper bounds on $R_<(H<, K{<(3)}_3(n))$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Theorem (thm-ordRamHyper-3UnifMaxDegLower), subsection “Bounded degrees and interval chromatic number”