Monochromatic spanning trees and matchings in ordered complete graphs

Abstract

AbstractWe study two well‐known Ramsey‐type problems for (vertex‐)ordered complete graphs. Two independent edges in ordered graphs can be nested, crossing, or separated. Apart from two trivial cases, these relations define six types of subgraphs, depending on which one (or two) of these relations are forbidden. Our first target is to refine a remark by Erdős 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 (nonnested, noncrossing, and nonseparated) 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, and crossing) subtrees of size only in a 2‐colored ordered complete graph on vertices. Some of these results relate to drawings of complete graphs. For instance, the existence of a monochromatic nonnested 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, that is, pairwise independent edges for ordered complete graphs. Cockayne and Lorimer proved that for given positive integers , is the smallest integer with the following property: every ‐coloring of the edges of a complete graph contains a monochromatic matching with edges. We conjecture that this result can be strengthened: ‐colored ordered complete graphs on vertices contain monochromatic nonnested and also nonseparated matchings with edges. We prove this conjecture for some special cases including the following. (i) Every ‐colored ordered complete graph on vertices contains a monochromatic nonnested matching of size two ( case). We prove it by showing that the chromatic number of the subgraph of the Kneser graph induced by the nonnested 2‐matchings in an ordered complete graph on vertices is ‐chromatic. (ii) Every 2‐colored ordered complete graph on vertices contains a monochromatic nonseparated matching of size ( case). This is the hypergraph analog of a result of Kaiser and Stehlík who proved that the Kneser graph induced by the nonseparated 2‐matchings in an ordered complete graph on vertices is ‐chromatic.For nested, separated, and crossing matchings the situation is different. The smallest ensuring a monochromatic matching of size in every ‐coloring is in the first two cases and one less in the third case.