The largest family of subsets of [n] without s pairwise disjoint sets is determined exactly when n = ms + c and s is sufficiently large.
Frankl,Proof of the Erd˝ os matching conjecture in a new range, Isr
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Families without $s$-matchings: the other end
The largest family of subsets of [n] without s pairwise disjoint sets is determined exactly when n = ms + c and s is sufficiently large.