For fixed m≥3 and c in the range β_m s^{(m-1)/m} to δ_m s, the extremal families avoiding s disjoint sets are the m-subsets of an (mℓ-1)-set union all sets of size at least m+1.
Frankl, Proof of the Erdős Matching Conjecture in a new range,Israel J
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Towards the Erd\H{o}s--Kleitman Problem: from Erd\H{o}s matching conjecture perspective
For fixed m≥3 and c in the range β_m s^{(m-1)/m} to δ_m s, the extremal families avoiding s disjoint sets are the m-subsets of an (mℓ-1)-set union all sets of size at least m+1.