On off-diagonal ordered Ramsey numbers of nested matchings

Abstract

For two graphs G< and H< with linearly ordered vertex sets, the ordered Ramsey numberr<(G<,H<) is the minimum N such that every red-blue coloring of the edges of the ordered complete graph on N vertices contains a red copy of G< or a blue copy of H<.For a positive integer n, a nested matchingNMn< is the ordered graph on 2n vertices with edges {i,2n−i+1} for every i=1,…,n. We improve bounds on the ordered Ramsey numbers r<(NMn<,K3<) obtained by Rohatgi, we disprove his conjecture by showing 4n+1≤r<(NMn<,K3<)≤(3+5)n+1 for every n≥6, and we determine the numbers r<(NMn<,K3<) exactly for n=4,5. As a corollary, this gives stronger lower bounds on the maximum chromatic number of k-queue graphs for every k≥3. We also prove r<(NMm<,Kn<)=Θ(mn) for arbitrary m and n.We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize all connected ordered graphs that are n-good for every n∈N. In particular, we discover a new class of ordered trees that are n-good for every n∈N, extending all the previously known examples.