Color-avoiding directed paths in tournaments
Jacob Fox, Benny Sudakov, and Yuval Wigderson

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 -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 is fixed and is sufficiently large, then every -edge-colored -vertex tournament contains a color-avoiding directed path of length . This answers a question of Gowers and Long, strengthens several of their results, and extends earlier work of Loh.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Computational Geometry and Mesh Generation · Advanced Graph Theory Research
