Random ordered matching versus triangle

Determine the growth rate of R_<(M^<(π),K^<_3) for random ordered matchings M^<(π) on n vertices with interval chromatic number 2.

Background

For random ordered matchings with interval chromatic number 2, the survey records upper and lower bounds that differ by roughly a factor of √n. The problem asks for the correct asymptotic growth rate.

References

What is the growth rate of $R_<(M<(\pi),K<_3)$ for the random ordered matchings $M<(\pi)$ on $n$ vertices with interval chromatic number 2?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem prob-random, Section 2.2

Even in the case of $M< (\pi)$ the bounds are roughly by the factor $\sqrt{n}$ apart, so Balko and Poljak asked the following question.

What is the growth rate of $R_<(M<(\pi),K<_3)$ for the random ordered matchings $M<(\pi)$ on $n$ vertices with interval chromatic number 2?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 2, subsection “Off-diagonal Ordered Ramsey Numbers,” Problem cited as [balPol24]

Even in the case of $M<(\pi)$ the bounds are roughly by the factor $\sqrt{n}$ apart, so Balko and Poljak asked the following question.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem~\ref{prob-random}, Subsection “Off-diagonal Ordered Ramsey Numbers”

The gap between our lower bound Ω(n{4/3}/(\log n){1/3}) and the upper bound O(n{7/4}) of Balko and Poljak remains open.

— Ordered matchings versus triangles via pseudorandom triangle-free graphs  (2609.19632 - Chen et al., 17 Sep 2026) in Concluding remarks; see also Problem 1 (Balko and Poljak, Problem 8)