On the sum of sizes of overlapping families

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 $n\geq 4k^2s$.