AI 中文总结
本文解决了极值组合学中竞赛图最大化有向圈的剩余情况，证明当ℓ被4整除时，旋转木马竞赛图的有向ℓ-圈数量渐近最多，且多于随机竞赛图，证实了Bartley和Day的猜想。
AI 中文摘要
确定能最大化指定子结构数量的组合结构是极值组合学的核心主题之一。Grzesik、Král'、Lovász和Volec证明，当ℓ不能被4整除时，在所有n顶点竞赛图中，随机竞赛图包含的长度为ℓ的有向圈数量渐近最多。本文解决了ℓ能被4整除的剩余情况，证明在此情况下，所谓的旋转木马竞赛图（carousel tournament）在所有n顶点竞赛图中渐近最大化了有向ℓ-圈的数量，且其包含的此类圈严格多于随机竞赛图，这证实了Bartley和Day的猜想。
英文摘要
Determining the combinatorial structures that maximize the number of prescribed substructures is a central theme in extremal combinatorics. Grzesik, Král', Lovász and Volec showed that when $\ell$ is not divisible by $4$, the random tournament contains asymptotically the most directed cycles of length $\ell$ among all $n$-vertex tournaments. In the paper, we resolve the remaining cases where $\ell$ is divisible by $4$. We show that, in this regime, the so-called carousel tournament asymptotically maximizes the number of directed $\ell$-cycles among all $n$-vertex tournaments, and in particular contains strictly more such cycles than the random tournament. This confirms the conjecture of Bartley and Day.
