Acyclic subgraphs of tournaments with high chromatic number

TL;DR

This paper investigates the chromatic number of acyclic subgraphs in tournaments, establishing bounds that confirm a conjecture and introduce new probabilistic and spectral techniques for analyzing tournament structures.

Contribution

It proves a strong form of a conjecture on acyclic subgraphs' chromatic number and introduces novel probabilistic and spectral methods for tournament analysis.

Findings

Every n-vertex tournament has an acyclic subgraph with chromatic number at least n^{5/9-o(1)}.

Existence of tournaments where all acyclic subgraphs have chromatic number at most n^{3/4+o(1)}.

New lemma linking many transitive subtournaments to large almost transitive subtournaments.

Abstract

We prove that every $n$-vertex tournament $G$ has an acyclic subgraph with chromatic number at least $n^{5/9-o(1)}$, while there exists an $n$-vertex tournament $G$ whose every acyclic subgraph has chromatic number at most $n^{3/4+o(1)}$. This establishes in a strong form a conjecture of Nassar and Yuster and improves on another result of theirs. Our proof combines probabilistic and spectral techniques together with some additional ideas. In particular, we prove a lemma showing that every tournament with many transitive subtournaments has a large subtournament that is almost transitive. This may be of independent interest.