Leading constant for multicolor bipartite ordered matchings

Determine the constant c such that R_<(M^<;q)=n^{cq+o(1)} for every ordered matching M^< on n vertices with interval chromatic number 2 and every integer q≥3.

Background

For interval-chromatic-two ordered matchings, the survey reports a growth rate n{Θ(q)}. The stated problem asks for the leading constant in the exponent.

References

It would be interesting to determine the leading constant in the exponent.

For any integer $q \geq 3$, determine the constant $c$ such that $R_<(M<; q) = n{c q + o(1)}$ for any ordered matching $M<$ on $n$ vertices with interval chromatic number two.

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

It would be interesting to determine the leading constant in the exponent.

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

For any integer $q \geq 3$, determine the constant $c$ such that $R_<(M<; q) = n{c q + o(1)}$ for any ordered matching $M<$ on $n$ vertices with interval chromatic number two.

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