Multicolor ordered matching upper bound

Show that, for every integer q≥3, there exists a constant c_q such that every ordered matching M^< on n vertices satisfies R_<(M^<;q)≤n^{c_q log n}.

Background

The known q-color upper bound for ordered matchings is n{(2 log n){q-1}}, and the cited authors believe a much stronger bound should hold. The problem proposes an upper bound with exponent proportional to log n for each fixed number of colors.

References

They believe that a much stronger upper bound should hold and pose the following problem.

For any integer $q \geq 3$, show that there exists a constant $c_q$ such that $R_<(M<; q) \leq n{c_q \log n}$ for any ordered matching $M<$ on $n$ vertices.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following multicolor ordered matching bounds, Section 2.5