  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 $[n] =
\{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most
${n-1\choose k-1}$.
  We say that two non-empty families are $\mathcal A, \mathcal B\subset
{[n]\choose k}$ are {\it $s$-cross-intersecting}, if for any $A\in\mathcal
A,B\in \mathcal B$ we have $|A\cap B|\ge s$. In this paper we determine the
maximum of $|\mathcal A|+|\mathcal B|$ for all $n$. This generalizes a result
of Hilton and Milner, who determined the maximum of $|\mathcal A|+|\mathcal B|$
for nonempty $1$-cross-intersecting families.
