Comparison of two-color and three-color off-diagonal thresholds

Determine whether there are ordered matchings M^< on n vertices for which the ratio R_<(K^<_3,K^<_3,M^<)/R_<(K^<_3,M^<) tends to infinity as n tends to infinity.

Background

The survey describes a multicolor analogue of a conjecture concerning whether adding another forbidden monochromatic triangle can substantially increase the Ramsey threshold for an ordered matching. The question asks whether the ratio of the three-color and two-color quantities can diverge.

References

Are there ordered matchings $M<$ on $n$ vertices such that $$\lim_{n \to \infty} \frac{R_<(K<_3, K<_3, M<)}{R_<(K<_3, M<)} = \infty?$$

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