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Directed paths: from Ramsey to Ruzsa and Szemeredi

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journal contribution
posted on 25.10.2015 by Po-Shen Loh

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-Szemeredi 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 n−−√⋅elog∗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-Szemeredi problem which we encounter through our investigation.

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25/10/2015

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