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.

中文翻译：

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色数较高的锦标赛的非循环子图

我们证明 $ñ$顶点比赛 $G$ 具有至少一个色数的无环子图 ${ñ}^{5/9-Ø（\mathrm{1个}）}$，尽管存在 $ñ$顶点比赛 $G$ 其每个无环子图最多具有色数 ${ñ}^{3/4+Ø（\mathrm{1个}）}$。这以强有力的形式确立了纳萨尔和尤斯特的猜想，并改善了他们的另一个结果。我们的证明结合了概率技术和频谱技术以及一些其他想法。尤其是，我们证明了一个引理，表明每一个具有很多传递子锦标赛的锦标赛都有一个几乎是传递的大型子锦标赛。这可能是独立利益。