A Product Version of the Hilton-Milner Theorem

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\dots ,n\}$ are called non-trivial cross-intersecting if $F\cap G\mathrm{\ne }\mathrm{\varnothing }$ for all $F\in \mathcal{F},G\in \mathcal{G}$ and $\cap \{F\text{\hspace{0.17em}⁣}:F\in \mathcal{F}\}=\mathrm{\varnothing }=\cap \{G\text{\hspace{0.17em}⁣}:G\in \mathcal{G}\}$. 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.