On off-diagonal ordered Ramsey numbers of nested matchings
Martin Balko, Marian Poljak

TL;DR
This paper advances understanding of ordered Ramsey numbers for nested matchings and complete graphs, disproves a conjecture, and explores ordered Ramsey goodness, revealing new classes of ordered trees with universal properties.
Contribution
It improves bounds on ordered Ramsey numbers for nested matchings and triangles, disproves Rohatgi's conjecture, and characterizes ordered trees that are n-good for all n.
Findings
Bounds on r_<(NM^<_n,K^<_3) are tightened, disproving Rohatgi's conjecture.
Exact values of r_<(NM^<_n,K^<_3) are determined for n=4,5.
New classes of ordered trees are identified as n-good for all n.
Abstract
For two graphs and with linearly ordered vertex sets, the ordered Ramsey number is the minimum such that every red-blue coloring of the edges of the ordered complete graph on vertices contains a red copy of or a blue copy of . For a positive integer , a nested matching is the ordered graph on vertices with edges for every . We improve bounds on the ordered Ramsey numbers obtained by Rohatgi, we disprove his conjecture by showing for every , and we determine the numbers exactly for . As a corollary, this gives stronger lower bounds on the maximum chromatic number of -queue graphs for every . We also prove for arbitrary and . We expand the…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Advanced Topology and Set Theory
