About sunflowers

Alon, Shpilka and Umans considered the following version of usual sunflower-free subset: a subset \mbox{\cal F}\subseteq \{1,\ldots ,D\}^n for $D>2$ is sunflower-free if for every distinct triple x,y,z\in \mbox{\cal F} there exists a coordinate $i$ where exactly two of $x_i,y_i,z_i$ are equal. Combining the polynomial method with character theory Naslund and Sawin proved that any sunflower-free set \mbox{\cal F}\subseteq \{1,\ldots ,D\}^n has size |\mbox{$\cal F$}|\leq c_D^n, where $c_D=\frac{3}{2^{2/3}}(D-1)^{2/3}$. In this short note we give a new upper bound for the size of sunflower-free subsets of $\{1,\ldots ,D\}^n$. Our main result is a new upper bound for the size of sunflower-free $k$-uniform subsets. More precisely, let $k$ be an arbitrary integer. Let \mbox{\cal F} be a sunflower-free $k$-uniform set system. Consider M:=|\bigcup\limits_{F\in \mbox{\cal F}} F|. Then |\mbox{$\cal F$}|\leq 3(\lceil\frac{2k}{3}\rceil+1)(2^{1/3}\cdot 3e)^k(\lceil\frac Mk\rceil -1)^{\lceil\frac{2k}{3}\rceil}. In the proof we use Naslund and Sawin's result about sunflower-free subsets in $\{1,\ldots ,D\}^n$.