Off-diagonal ordered Ramsey numbers of matchings

Abstract

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, 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 $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random matchings with interval chromatic number $2$.