The maximum number of S_{r-1,k}^r copies in an r-uniform hypergraph with matching number at most ν is independent of k and equals the number in the extremal construction given by the Erdős Matching Conjecture; this implies the conjecture holds in the (r-1,k)-norm for all k.
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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.
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.
Proves an approximate form of the conjecture on e(s(m+1)-ℓ, s) for ℓ ≤ s/2 when s ≥ s0(m).
The extremal number ex(n, {M^r_{s+1}, K_{ℓ+1}^{(r)+}}) is determined exactly for s small or sufficiently large, via a new rainbow hyper-Turán result for clique expansions.
The largest family of subsets of [n] without s pairwise disjoint sets is determined exactly when n = ms + c and s is sufficiently large.
citing papers explorer
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Counting sunflowers in hypergraphs with bounded matching number and Erd\H{o}s Matching Conjecture in the $(t,k)$-norm
The maximum number of S_{r-1,k}^r copies in an r-uniform hypergraph with matching number at most ν is independent of k and equals the number in the extremal construction given by the Erdős Matching Conjecture; this implies the conjecture holds in the (r-1,k)-norm for all k.
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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.
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More on the Erd\H os--Kleitman problem on matchings in set families
Proves an approximate form of the conjecture on e(s(m+1)-ℓ, s) for ℓ ≤ s/2 when s ≥ s0(m).
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Hypergraph extensions of the Alon--Frankl Theorem and rainbow Tur\'an problems
The extremal number ex(n, {M^r_{s+1}, K_{ℓ+1}^{(r)+}}) is determined exactly for s small or sufficiently large, via a new rainbow hyper-Turán result for clique expansions.
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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.