  In this paper, we show that, given two down-sets (simplicial complexes) there
is a matching between them that matches disjoint sets and covers the smaller of
the two down-sets. This result generalizes an unpublished result of Berge from
circa 1980. The result has nice corollaries for cross-intersecting families and
Chv\'atal's conjecture. More concretely, we show that Chv\'atal's conjecture is
true for intersecting families with covering number $2$.
  A family $\mathcal F\subset 2^{[n]}$ is intersection-union (IU) if for any
$A,B\in\mathcal F$ we have $1\le |A\cap B|\le n-1$. Using the aforementioned
result, we derive several exact product- and sum-type results for IU-families.
