Color-avoiding directed paths in tournaments

TL;DR

This paper proves that in large enough q-colored tournaments, there exists a long directed path avoiding one color, with length close to the total number of vertices, answering a question posed by Gowers and Long.

Contribution

It establishes a lower bound on the length of color-avoiding directed paths in q-colored tournaments for large q, extending previous results and answering an open question.

Findings

Existence of long color-avoiding paths in large tournaments

Improved bounds on path length relative to tournament size

Extension of prior work by Loh and others

Abstract

We study the following Ramsey-theoretic question: given a $q$-coloring of the edges of a tournament, how long of a directed path can we guarantee whose edges avoid one of the colors? Questions of this type have applications in many areas, such as vector sequences, convex geometry, and extremal hypergraph theory, and have been extensively studied over the past 50 years. We prove that if $\varepsilon>0$ is fixed and $q$ is sufficiently large, then every $q$-edge-colored $N$-vertex tournament contains a color-avoiding directed path of length $N^{1-\varepsilon}$. This answers a question of Gowers and Long, strengthens several of their results, and extends earlier work of Loh.