For the Erdős–Kleitman problem e(ms+c,s), specific families P' are proven extremal for β_m s^{(m-1)/m} ≤ c ≤ δ_m s (m fixed), with a sharpened range for m=3, and a new family R disproves the Kupavskii–Sokolov conjecture in an intermediate range.
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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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New Extremal Ranges and Constructions of the Erd\H{o}s--Kleitman Problem
For the Erdős–Kleitman problem e(ms+c,s), specific families P' are proven extremal for β_m s^{(m-1)/m} ≤ c ≤ δ_m s (m fixed), with a sharpened range for m=3, and a new family R disproves the Kupavskii–Sokolov conjecture in an intermediate range.
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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.