Monochromatic spanning trees and matchings in ordered complete graphs
J\'anos Bar\'at, Andr\'as Gy\'arf\'as, G\'eza T\'oth

TL;DR
This paper investigates monochromatic spanning trees and matchings in ordered complete graphs, establishing conditions under which certain monochromatic subgraphs exist based on forbidden relations and coloring constraints.
Contribution
It refines known Ramsey results for ordered graphs, proving the existence of specific monochromatic spanning trees and matchings under various forbidden relation conditions.
Findings
Monochromatic non-nested spanning trees always exist in 2-colorings when one relation is forbidden.
Existence of monochromatic matchings of size n in ordered graphs is confirmed for certain cases.
Thresholds for the smallest number of vertices ensuring monochromatic matchings are established.
Abstract
Two independent edges in ordered graphs can be nested, crossing or separated. These relations define six types of subgraphs, depending on which relations are forbidden. We refine a remark by Erd\H{o}s and Rado that every 2-coloring of the edges of a complete graph contains a monochromatic spanning tree. We show that forbidding one relation we always have a monochromatic (non-nested, non-crossing, non-separated) spanning tree in a 2-edge-colored ordered complete graph. On the other hand, if two relations are forbidden, then it is possible that we have monochromatic (nested, separated, crossing) subtrees of size only half the number of vertices. The existence of a monochromatic non-nested spanning tree in 2-colorings of ordered complete graphs verifies a more general conjecture for twisted drawings. Our second subject is to refine the Ramsey number of matchings for ordered complete…
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