  We combine here Tao's slice-rank bounding method and Gr\"obner basis
techniques and apply here to the Erd\H{o}s-Rado Sunflower Conjecture.
  Let $\frac{3k}{2}\leq n\leq 3k$ be integers. We prove that if $\mbox{$\cal
F$}$ be a $k$-uniform family of subsets of $[n]$ without a sunflower with 3
petals, then $$ |\mbox{$\cal F$}|\leq 3{n \choose n/3}. $$
  We give also some new upper bounds for the size of a sunflower-free family in
$2^{[n]}$.
