A Strengthening of the Erd\H{o}s-Szekeres Theorem
@article{Balogh2020ASO,
title={A Strengthening of the Erd\H\{o\}s-Szekeres Theorem},
author={J{\'o}zsef Balogh and Felix Christian Clemen and Emily Heath and Mikhail Lavrov},
journal={arXiv: Combinatorics},
year={2020}
}The Erdős-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ 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 $K_{rs+1}$ are necessary. We prove the following surprising characterization of ordered graphs on $rs+1$ vertices with this coloring…
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