TY - JOUR

T1 - On off-diagonal ordered Ramsey numbers of nested matchings

AU - Balko, Martin

AU - Poljak, Marian

N1 - Funding Information:
Martin Balko was supported by the grant no. 19-04113Y of the Czech Science Foundation (GAČR) and by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). This article is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 810115).
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2023/2/1

Y1 - 2023/2/1

N2 - 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+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.

AB - 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+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.

KW - Nested matching

KW - Ordered graph

KW - Ordered Ramsey number

KW - Ramsey goodness

UR - http://www.scopus.com/inward/record.url?scp=85141336473&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2022.113223

DO - 10.1016/j.disc.2022.113223

M3 - Article

AN - SCOPUS:85141336473

VL - 346

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2

M1 - 113223

ER -