A strengthening of the Erdős–Szekeres Theorem

József Balogh, University of Illinois Urbana-Champaign
Felix Christian Clemen, University of Illinois Urbana-Champaign
Emily Heath, Iowa State University
Mikhail Lavrov, Kennesaw State University


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.