Families without $s$-matchings: the other end
This solves a long-standing open problem in extremal set theory for a specific parameter range, providing a non-uniform analogue of the Erdős Matching Conjecture.
The paper determines the largest family of subsets of an n-element set without s pairwise disjoint sets, for n=ms+c and sufficiently large s, extending the Erdős Matching Conjecture to non-uniform families in the regime where the extremal construction is a clique.
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 $s \geq s_0(m, c)$. 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.