  Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an
$n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$
distinct families. Subject to this condition we investigate the maximum of
$|\mathcal{A}_1|+\ldots+|\mathcal{A}_m|$. Note that the subcase $m=s+1$,
$\mathcal{A}_1=\ldots=\mathcal{A}_m$ is the Erd\H{o}s Matching Conjecture, one
of the most important open problems in extremal set theory. We provide some
upper bounds, a general conjecture and its solution for the range $n\geq
4k^2s$.
