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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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.