A strengthening of the Erdős–Szekeres Theorem. (March 2022) - Balogh, József
Abstract: The Erdős–Szekeres Theorem stated in terms of graphs says that any red–blue coloring of the edges of the ordered complete graph K r s + 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 r s + 1 is the minimum number of vertices needed for this result, not all edges of K r s + 1 are necessary. We characterize the subgraphs of K r s + 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 C T ( 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.