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arXiv · 1912.07722

Acyclic subgraphs of tournaments with high chromatic number

Jacob Fox·Matthew Kwan·Benny Sudakov
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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.

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Jacob Fox, Matthew Kwan, Benny Sudakov. 2019-12-16. Acyclic subgraphs of tournaments with high chromatic number. https://doi.org/10.1112/blms.12446

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