Counting sunflowers with restricted matching number

Abstract

For a family $\mathcal{H}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$, a subset $\{A_1,A_2,\dots ,A_m\}\subseteq \mathcal{H}$ is called a \textit{matching} of size~$m$ if the sets $A_1,A_2,\dots ,A_m$ are pairwise disjoint. The \textit{matching number} of $\mathcal{H}$, denoted by $\hat{I}½(\mathcal{H})$, is the largest integer~$m$ for which such a matching exists. $\{A_1,A_2,\dots ,A_l\}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ 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 $1\le i\le l$. 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 $c{o}_p(\mathcal{H})={\sum }_{E\in \left(\genfrac{}{}{0ex}{}{[n]}{k-1}\right)}(d_{\mathcal{H}}(E){)}^p$. 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 $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ with matching number $\hat{I}½(\mathcal{F})=s$. These results can be viewed as a Turán-type problem (specifically ${\mathrm{ex}}_k(n,S_{k,l}^{k-1},M_s)$) and a generalization of the Erdős Matching Conjecture. Furthermore, for the case $k=3$, we establish a linear threshold for $n$.

17 pages