A Product Version of the Hilton-Milner-Frankl Theorem

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\dots ,n\}$ are called non-trivial cross $t$-intersecting if $\mathrm{\mid }F\cap G\mathrm{\mid }\ge t$ for all $F\in \mathcal{F},G\in \mathcal{G}$ and $\mathrm{\mid }\cap \{F\text{\hspace{0.17em}⁣}:F\in \mathcal{F}\}\mathrm{\mid }<t$, $\mathrm{\mid }\cap \{G\text{\hspace{0.17em}⁣}:G\in \mathcal{G}\}\mathrm{\mid }<t$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross $t$-intersecting families of $k$-subsets of $\{1,2,\dots ,n\}$ for $n\ge 4(t+2{)}^2{k}^2$, $k\ge 5$, which is a product version of the Hilton-Milner-Frankl Theorem.