A strengthening of the Erdős–Szekeres Theorem

József Balogh, Felix Christian Clemen, Emily Heath, Mikhail Lavrov

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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.

Original languageEnglish (US)
Article number103456
JournalEuropean Journal of Combinatorics
StatePublished - Mar 2022

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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