Ordered Ramsey numbers of matchings

Close the gap between the upper and lower bounds for ordered Ramsey numbers of matchings.

Background

The survey reports nearly matching polynomial upper and superpolynomial lower bounds for ordered matchings, but notes a remaining gap in the exponent. The cited problem asks for sharper quantitative bounds in this basic sparse class.

References

Conlon, Fox, Lee, and Sudakov thus posed the following problem.

Close the gap between the upper and lower bounds for ordered Ramsey numbers of matchings.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Equation (eq-ordRam-degInt), Section 2

Although the bounds are quite close, there is still a small gap in the exponent, even for $d=1$. Conlon, Fox, Lee, and Sudakov thus posed the following problem.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following equation (in Section 2, Subsection “Bounded degrees and interval chromatic number”)

Close the gap between the upper and lower bounds for ordered Ramsey numbers of matchings.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Equation (eq-ordRam-degInt), Section 2, subsection “Bounded degrees and interval chromatic number”