A Strengthening of the Erd\H{o}s-Szekeres Theorem

TL;DR

This paper strengthens the Erd ext{"o}s-Szekeres Theorem by characterizing minimal subgraphs that guarantee a monochromatic monotone path in edge-colored complete graphs and improves bounds on related Ramsey numbers.

Contribution

It characterizes subgraphs with the coloring property using the novel circus tent graph and improves bounds on the online ordered size Ramsey number of a path.

Findings

Characterization of subgraphs containing all edges of the circus tent graph CT(r,s).

Identification of minimal subgraphs guaranteeing monochromatic paths.

Improved bounds on the online ordered size Ramsey number of paths.

Abstract

The Erd\H{o}s-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone increasing path with $s$ edges. Although $rs + 1$ is the minimum number of vertices needed for this result, not all edges of $K_{rs+1}$ are necessary. We characterize the subgraphs of $K_{rs+1}$ with this coloring property as follows: they are exactly the subgraphs that contain all the edges of a graph we call the circus tent graph $CT(r,s)$. Additionally, we use similar proof techniques to improve upon some of the bounds on the online ordered size Ramsey number of a path given by P\'erez-Gim\'enez, Pralat, and West.