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arXiv · 2006.03703

A Strengthening of the Erd\H{o}s-Szekeres Theorem

József Balogh·Felix Christian Clemen·Emily Heath·Mikhail Lavrov
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Abstract

The Erd\H{o}s-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 characterize the subgraphs of $K_{rs+1}$ 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 some of the bounds on the online ordered size Ramsey number of a path given by P\'erez-Gim\'enez, Pralat, and West.

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József Balogh, Felix Christian Clemen, Emily Heath, Mikhail Lavrov. 2020-06-05. A Strengthening of the Erd\H{o}s-Szekeres Theorem. https://arxiv.org/abs/2006.03703

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