If $2\le d\le k$ and $n\ge dk/(d-1)$, a d-cluster is defined to be a collection of d elements of $\left(\begin{array}{c}[n]\\ k\end{array}\right)$ with empty intersection and union of size no more than 2k. Mubayi [14] conjectured that the largest size of a d-cluster-free family $\mathcal{F}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$ is $\left(\begin{array}{c}n-1\\ k-1\end{array}\right)$, with equality holding only for a maximum-sized star. Here, we resolve Mubayi's conjecture and prove a slightly stronger result, thus completing a new generalization of the Erdős-Ko-Rado Theorem.

中文翻译：

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关于Erdős-Ko-Rado的d聚类推广

如果 $\mathrm{2个}\le d\le ķ$ 和 $ñ\ge dķ/（d-\mathrm{1个}）$，则d -cluster定义为的d个元素的集合$（\begin{array}{c}[ñ]\\ ķ\end{array}）$空交点和并集的大小不超过2 k。Mubayi [14]推测，无d族的家庭最大$\mathcal{F}\subset （\begin{array}{c}[ñ]\\ ķ\end{array}）$ 是 $（\begin{array}{c}ñ-\mathrm{1个}\\ ķ-\mathrm{1个}\end{array}）$，而等式仅适用于最大尺寸的恒星。在这里，我们解决了Mubayi的猜想并证明了稍微强一些的结果，从而完成了Erdős-Ko-Rado定理的新推广。