Counting sunflowers with restricted matching number

Abstract

For H ⊆ ( [ n ] k ) , let ν ( H ) denote its matching number and let d H ( E ) denote the codegree of a ( k − 1 ) -set E . We study the codegree moments c o p ( H ) = ∑ E ∈ ( [ n ] k − 1 ) d H ( E ) p and the number of copies of the k -uniform l -petal sunflower S k , l k − 1 whose core has size k − 1 . For fixed positive integers k , l ≥ 2 and s , p ≥ 1 , and all sufficiently large n , we prove that among all k -uniform families with matching number at most s , both quantities are uniquely maximized, up to permutation, by H n , k , s = { F ∈ ( [ n ] k ) : F ∩ [ s ] ≠ ∅ } . This determines the generalized Turán number ex k ( n , S k , l k − 1 , M s + 1 ) , where M s + 1 consists of s + 1 pairwise disjoint edges. When k = 3 , the conclusion for codegree moments holds for n ≥ 6 s + 3 , uniformly for all positive integers p . The proof combines a theorem of Frankl and Kupavskii on families with bounded matching number, an analysis of ( k − 1 ) -sets of large codegree, and a moment inequality.