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

TL;DR

This paper explores the connections between directed paths in tournaments and the Ruzsa-Szemerédi problem, establishing new bounds and generalizations through combinatorial and Ramsey-theoretic methods.

Contribution

It introduces novel links between directed paths and the Ruzsa-Szemerédi problem, proving new bounds for colored tournaments and exploring related constructions and generalizations.

Findings

Every 3-coloring of a transitive tournament contains a long directed path avoiding one color.

Established a lower bound of rom the abstract, likely or the length of such paths.

Connected the problem to the Ruzsa-Szemerdi induced matching problem.

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.