Lower bounds for regular ordered graphs with interval chromatic number two

Construct, for every fixed maximum degree Δ, a sequence of n-vertex Δ-regular ordered graphs with interval chromatic number two whose ordered Ramsey numbers are at least n^{cΔ} for some absolute constant c>0.

Background

Polynomial upper bounds are known for bounded-degree ordered graphs with bounded interval chromatic number, but the survey states that no nontrivial lower bounds were known for this setting. The problem seeks regular examples exhibiting quantitatively large ordered Ramsey numbers.

References

However, we have no non-trivial lower bounds for this case and thus Balko, Cibulka, Kr {a}l, and Kyn\v{c}l stated the following problem.

Is there a constant $c>0$ such that for every fixed $\Delta$ there is a sequence ${G<n}{n\in\mathbb{N}$ of ordered $\Delta$-regular graphs $G<_n$ with $n$ vertices and interval chromatic number $2$ such that $R_<(G<_n) \ge n{c\Delta}$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 2, subsection “Bounded degrees and interval chromatic number,” Problem cited as [bckk13]

However, we have no non-trivial lower bounds for this case and thus Balko, Cibulka, Kr{a}l, and Kyn\v{c}l stated the following problem.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following equation (in Section 2, Subsection “Bounded degrees and interval chromatic number”)

Is there a constant $c>0$ such that for every fixed $\Delta$ there is a sequence ${G<n}{n\in\mathbb{N}$ of ordered $\Delta$-regular graphs $G<_n$ with $n$ vertices and interval chromatic number $2$ such that $R_<(G<_n) \ge n{c\Delta}$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Equation (eq-ordRam-degInt), Section 2, subsection “Bounded degrees and interval chromatic number”