For fixed m≥3 and large s, the extremal families achieving e((m+1)s−ℓ,s) are exactly the P(m,s,ℓ;L) families when 1≤ℓ≤((m+1)/(2m+1)−o(1))s, confirming the Frankl-Kupavskii conjecture in this regime.
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A solution to Frankl and Kupavskii's conjecture concerning Erd\H{o}s-Kleitman matching problem
For fixed m≥3 and large s, the extremal families achieving e((m+1)s−ℓ,s) are exactly the P(m,s,ℓ;L) families when 1≤ℓ≤((m+1)/(2m+1)−o(1))s, confirming the Frankl-Kupavskii conjecture in this regime.