On off-diagonal ordered Ramsey numbers of nested matchings

For two graphs G^< and H^< with linearly ordered vertex sets, the ordered Ramsey number r_<(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 matching NM_n^< 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_<(NM_n^<,K₃^<) obtained by Rohatgi, we disprove his conjecture by showing 4n+1≤r_<(NM_n^<,K₃^<)≤(3+5)n+1 for every n≥6, and we determine the numbers r_<(NM_n^<,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_<(NM_m^<,K_n^<)=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.