Counting sunflowers with restricted matching number

Abstract

For a family $\mathcal{H} \subseteq \binom{[n]}{k}$, a subset $\{A_1, A_2, \ldots, A_m\} \subseteq \mathcal{H}$ is called a \textit{matching} of size~$m$ if the sets $A_1, A_2, \ldots, A_m$ are pairwise disjoint. The \textit{matching number} of $\mathcal{H}$, denoted by $\nu(\mathcal{H})$, is the largest integer~$m$ for which such a matching exists. $\{A_1,A_2,\ldots,A_l\}\subseteq \binom{[n]}{k}$ is said to be a \textit{$k$-uniform sunflower} with $l$ \textit{petals}, if there exists a core set $C\subseteq[n]$ contained in every $A_i$ and $A_i\setminus C$ are pairwise disjoint, for $1\leq i\leq l$. Let $S_{k,l}^{k-1}$ denote the $k$-uniform sunflower with $l$ petals and the core set of size $k-1$. The \textit{codegree} of $E$ in $\mathcal{H}$, denoted by $d_{\mathcal{H}}(E)$, is defined as $d_{\mathcal{H}}(E) =|\{F\in \mathcal{H}:E\subseteq F\}|$. Let the \textit{$\ell_p$-norm} of $\mathcal{H}$ be $co_p(\mathcal{H})= \sum_{E\in \binom{[n]}{k-1}}(d_{\mathcal{H}}(E))^p$. For sufficiently large $n$, we determine the maximum $\ell_p$-norm and the maximum number of sunflowers $S_{k,l}^{k-1}$ for a family $\mathcal{F} \subseteq \binom{[n]}{k}$ with matching number $\nu(\mathcal{F}) = s$. These results can be viewed as a Tur\'an-type problem (specifically $\mathrm{ex}_k(n, S_{k,l}^{k-1}, M_s)$) and a generalization of the Erd\H{o}s Matching Conjecture. Furthermore, for the case $k = 3$, we establish a linear threshold for $n$.