Two families $\mathcal{F},\mathcal{G}$ of k-subsets of $\{1,2,\dots ,n\}$ are called non-trivial cross-intersecting if $F\cap G\ne \varnothing$ for all $F\in \mathcal{F},G\in \mathcal{G}$ and $\cap \{F:F\in \mathcal{F}\}=\varnothing =\cap \{G:G\in \mathcal{G}\}$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of k-subsets of $\{1,2,\dots ,n\}$ for $n\ge 4k$, $k\ge 8$, which is a product version of the classical Hilton-Milner Theorem.

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希尔顿米尔纳定理的产品版本

两个家庭$\mathcal{F},\mathcal{G}$k的子集$\{1,2,\dots \dots ,n\}$被称为非平凡交叉如果$F\cap G\ne \varnothing$对全部$F\epsilon \mathcal{F},G\epsilon \mathcal{G}$和$\cap \{F:F\epsilon \mathcal{F}\}=\varnothing =\cap \{G:G\epsilon \mathcal{G}\}$。在本文中，我们确定了k子集的两个非平凡交叉族的大小的最大乘积$\{1,2,\dots \dots ,n\}$为了$n\ge 4k$,$k\ge 8$，这是经典希尔顿米尔纳定理的产品版本。