Multicolor ordered Ramsey numbers of matchings

Prove that for every integer $q\ge3$ there exists a constant $c_q$ such that every ordered matching $M^<$ on $n$ vertices satisfies $R_<(M^<;q)\le n^{c_q\log n}$.

Background

The known upper bound for the qq-color ordered Ramsey number of an arbitrary ordered matching has an exponent involving (log⁡n)q−1(\log n)^{q-1}. The proposed problem seeks a substantially stronger bound with only one factor of log⁡n\log n in the exponent for each fixed number of colors.

References

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

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

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 Section 2, subsection “Multicolor Ordered Ramsey Numbers,” Problem cited as [clfs17]

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 in subsection “Multicolor Ordered Ramsey Numbers”