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.
More on the Erd\H os--Kleitman problem on matchings in set families
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. In 1968, answering a question of Erd\H{o}s, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integers $m,s\ge 1$. Half a century later, Frankl and Kupavskii determined $e(s(m+1)-\ell, s)$ for $\ell \leq \frac{s-3}{m+3}$. They showed that the corresponding extremal example is closely connected with the extremal example for the Erd\H{o}s Matching Conjecture, and conjectured that the same remains true for all $\ell \leq s/2$. In this paper, we prove an approximate version of their conjecture for $s\ge s_0(m)$.
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math.CO 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
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.
citing papers explorer
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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.
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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.