A new upper bound for the size of a sunflower-free family

Abstract: 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 $2^{[n]}$.