Minimum ordered Ramsey numbers of 3-regular graphs

Close the gap between the upper and lower bounds for minimum ordered Ramsey numbers of 3-regular graphs.

Background

For random 3-regular graphs, the survey gives a lower bound of order n{7/6}/(log n log log n) and an upper bound O(n9) for the minimum over all vertex orderings. The problem asks for substantially sharper bounds.

References

Since the gap is rather large, Balko, Jel{i}nek, and Valtr posed the following problem.

Close the gap between the upper and lower bounds for minimum ordered Ramsey numbers of 3-regular graphs.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Equation (eq-ordRam-minRamseyUpper), Section 2.3

Close the gap between the upper and lower bounds for minimum ordered Ramsey numbers of 3-regular graphs.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Equation (eq-ordRam-minRamseyUpper), subsection “Minimum Ordered Ramsey Numbers”