Two families $\mathcal{ F},\mathcal{ G}$ of $k$-subsets of $\{ 1,2,\ldots,n\}$ are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in \mathcal{ F}, G\in \mathcal{ G}$ and $|\cap \{ F\colon F\in \mathcal{ F}\}|<t$, $|\cap \{ G\colon G\in\mathcal{ G}\}|<t$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross $t$-intersecting families of $k$-subsets of $\{ 1,2,\ldots,n\}$ for $n\geq 4(t+2)^2k^2$, $k\geq 5$, which is a product version of the Hilton-Milner-Frankl Theorem.
Categories: math.CO