Let S be an n-element set and $\mathcal{F}\subset \left(\begin{array}{c}S\\ k\end{array}\right)$ an intersecting family. Improving earlier results it is proved that for $n>72k$ there is an element of S that is contained in all but $\left(\begin{array}{c}n-3\\ k-2\end{array}\right)$ members of $\mathcal{F}$. One of the main ingredients of the proof is the following statement. If $\mathcal{G}\subset \left(\begin{array}{c}S\\ k\end{array}\right)$ is intersecting, $|\mathcal{G}|\ge \left(\begin{array}{c}n-2\\ k-2\end{array}\right)$ and $n\ge 72k$ then there is an element of S that is contained in more than half of the members of $\mathcal{G}$.

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相交超图的最大程度和多样性

令S为n元素集，$\mathcal{F}\subset （\begin{array}{c}\mathrm{小号}\\ ķ\end{array}）$一个相交的家庭。改善早期结果可证明$ñ>72ķ$除了所有元素，都包含一个S元素$（\begin{array}{c}ñ-3\\ ķ-2\end{array}）$ 的成员 $\mathcal{F}$。证明的主要内容之一是以下陈述。如果$\mathcal{G}\subset （\begin{array}{c}\mathrm{小号}\\ ķ\end{array}）$ 相交， $|\mathcal{G}|\ge （\begin{array}{c}ñ-2\\ ķ-2\end{array}）$ 和 $ñ\ge 72ķ$则S的一半以上成员中包含S的元素$\mathcal{G}$。