Off-diagonal ordered Ramsey numbers of matchings
Dhruv Rohatgi

TL;DR
This paper investigates the off-diagonal ordered Ramsey numbers for matchings, providing new bounds and resolving special cases of a conjecture about their asymptotic behavior, especially for non-crossing and certain random matchings.
Contribution
The paper establishes nearly linear bounds for non-crossing matchings and sub-quadratic bounds for random matchings with interval chromatic number 2, advancing understanding of ordered Ramsey numbers.
Findings
Nearly linear bounds for non-crossing matchings
Sub-quadratic bounds for random matchings with interval chromatic number 2
Progress on conjecture about asymptotic bounds of ordered Ramsey numbers
Abstract
For ordered graphs and , the ordered Ramsey number is the smallest such that every red/blue edge coloring of the complete graph on vertices contains either a blue copy of or a red copy of , where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the "off-diagonal" ordered Ramsey number , where is an ordered matching on vertices. In particular, Conlon et al. asked what asymptotic bounds (in ) can be obtained for , where the maximum is over all ordered matchings on vertices. The best-known upper bound is , whereas the best-known lower bound is , and Conlon et al. hypothesize that for every ordered matching . We resolve two special…
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