A strengthening of the Erdős–Szekeres Theorem
Department
Mathematics
Document Type
Article
Publication Date
3-1-2022
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.
Journal Title
European Journal of Combinatorics
Journal ISSN
01956698
Volume
101
Digital Object Identifier (DOI)
10.1016/j.ejc.2021.103456