A Product Version of the Hilton-Milner-Frankl Theorem

Abstract

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

arXiv commentto appear in Science China Mathematics. arXiv admin note: text overlap with arXiv:2205.00109