A strengthening of the Erdős–Szekeres Theorem

Abstract

The Erdős–Szekeres Theorem stated in terms of graphs says that any red–blue coloring of the edges of the ordered complete graph Krs+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 Krs+1 are necessary. We characterize the subgraphs of Krs+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 the bounds on the online ordered size Ramsey number of a path given by Pérez-Giménez, Prałat, and West.