Counting sunflowers with restricted matching number

Abstract

For $\mathcal H\subseteq\binom{[n]}k$, let $\nu(\mathcal H)$ denote its matching number and let $d_{\mathcal H}(E)$ denote the codegree of a $(k-1)$-set $E$. We study the codegree moments $ co_p(\mathcal H)=\sum_{E\in\binom{[n]}{k-1}}d_{\mathcal 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 $k,s,l$ and $p$, and all sufficiently large $n$, we prove that among all $k$-uniform families with matching number at most $s$, both quantities are uniquely maximized by \[ \mathcal H_{n,k,s}=\left\{F\in\binom{[n]}k:F\cap[s]\neq\emptyset\right\}. \] Thus the sunflower result is the generalized Tur\'an problem $\operatorname{ex}_k(n,S_{k,l}^{k-1},M_{s+1})$. When $k=3$, we make the threshold effective: for every integer $p\geq1$, the codegree-moment conclusion holds for $n\geq6s+3$. The proof combines a Hilton--Milner-type stability theorem of Frankl and Kupavskii with a high-codegree-shadow reduction and a convex moment estimate.