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arXiv · 1505.07312

Directed paths: from Ramsey to Ruzsa and Szemer\'edi

Po-Shen Loh
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Abstract

Starting from an innocent Ramsey-theoretic question regarding directed paths in tournaments, we discover a series of rich and surprising connections that lead into the theory around a fundamental problem in Combinatorics: the Ruzsa-Szemer\'edi induced matching problem. Using these relationships, we prove that every coloring of the edges of the transitive $n$-vertex tournament using three colors contains a directed path of length at least $\sqrt{n} \cdot e^{\log^* n}$ which entirely avoids some color. We also expose connections to a family of constructions for Ramsey tournaments, and introduce and resolve some natural generalizations of the Ruzsa-Szemer\'edi problem which we encounter through our investigation.

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Po-Shen Loh. 2015-05-27. Directed paths: from Ramsey to Ruzsa and Szemer\'edi. https://arxiv.org/abs/1505.07312

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