Typical off-diagonal ordered matching bound

Prove that, for every positive integer χ, there exists ε(χ)>0 such that almost every ordered matching M^< on n vertices with interval chromatic number χ satisfies R_<(M^<,K^<_3) ≤ O(n^{2-ε(χ)}).

Background

The survey presents this as a conjecture for almost all ordered matchings with a fixed interval chromatic number. It is a probabilistic weakening of the problem asking for a subquadratic bound for every ordered matching.

References

For a positive integer $\chi$, there is a constant $\varepsilon(\chi)>0$ such that $R_<(M<,K<_3) \leq O(n{2-\varepsilon(\chi)})$ for almost every ordered matching $M<$ on $n$ vertices with interval chromatic number~$\chi$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 2, subsection “Off-diagonal Ordered Ramsey Numbers”