Counting sunflowers with restricted matching number

Abstract

For $\mathcal{H}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$, 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 $c{o}_p(\mathcal{H})={\sum }_{E\in \left(\genfrac{}{}{0ex}{}{[n]}{k-1}\right)}{d}_{\mathcal{H}}(E{)}^p$ 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 Turán problem ${\mathrm{ex}}_k(n,S_{k,l}^{k-1},M_{s+1})$. When $k=3$, we make the threshold effective: for every integer $p\ge 1$, the codegree-moment conclusion holds for $n\ge 6s+3$. The proof combines a Hilton--Milner-type stability theorem of Frankl and Kupavskii with a high-codegree-shadow reduction and a convex moment estimate.