Linial and Morgenstern conjectured that, among all n-vertex tournaments with $d\left(\begin{array}{c}n\\ 3\end{array}\right)$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for $d\ge 1/36$ by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate that the family of extremal examples is broader than expected and give its full description for $d\ge 1/16$.

中文翻译：

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比赛中长度为3和4的循环

Linial和Morgenstern推测，在所有n -vertex比赛中，$d（\begin{array}{c}ñ\\ 3\end{array}）$长度为3的循环，长度为4的循环数通过传递性锦标赛的随机爆炸而渐近最小化，传递中的所有部分（大小相等的一部分除外）均较小，而另一部分较小。我们证明了$d\ge \mathrm{1个}/36$通过分析锦标赛邻接矩阵的频谱。我们还证明了极端例子的家族比预期的要广泛，并提供了完整的描述$d\ge \mathrm{1个}/16$。