Uniform s-cross-intersecting families

In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado theorem, which states that any family of $k$-element subsets of the set $[n] = {1,\ldots,n}$inwhichanytwosetsintersect,hascardinalityatmost${n-1\choose k-1}$. We say that two non-empty families are $\mathcal A, \mathcal B\subset {[n]\choose k}are {\its-cross-intersecting}, if for anyA\in\mathcal A,B\in \mathcal B$wehave$|A\cap B|\ge s$. In this paper we determine the maximum of $\mathrm{\mid }\mathcal{A}\mathrm{\mid }+\mathrm{\mid }\mathcal{B}\mathrm{\mid }$ for all $n$. This generalizes a result of Hilton and Milner, who determined the maximum of $\mathrm{\mid }\mathcal{A}\mathrm{\mid }+\mathrm{\mid }\mathcal{B}\mathrm{\mid }$ for nonempty $1$-cross-intersecting families.