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 NMn< 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 + root 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< ) =O(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 is an element of N. In particular, we discover a new class of ordered trees that are n-good for every n is an element of N, extending all the previously known examples.(c) 2022 Elsevier B.V. All rights reserved.