Families without $s$-matchings: the other end

In this paper, we determine the largest family $\mathcal{F}\subset 2^{[n]}$ without $s$ pairwise disjoint sets, provided $n=ms+c$ for positive integers $m,c$, and $s\ge s_0(m,c)$. This result can be seen as a non-uniform analogue of the results on the Erd\H os Matching Conjecture in the regime when the clique is extremal.