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.
Erd˝ os, A problem on independentr-tuples,Ann
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For large n, any n-vertex r-uniform hypergraph with matching number < s has spectral radius at most that of F_{s-1}(n), with equality only for that hypergraph.
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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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A Spectral Confirmation of the Erd\H{o}s Matching Conjecture
For large n, any n-vertex r-uniform hypergraph with matching number < s has spectral radius at most that of F_{s-1}(n), with equality only for that hypergraph.