An improved upper bound for the size of a sunflower-free family

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We combine here Tao’s slice-rank bounding method and Grobner basis techniques and apply it to the Erdős–Rado Sunflower Conjecture. Let $${0\leq k\leq n}$$ be integers. We prove that if $${\mathcal{F}}$$ is a k-uniform family of subsets of [n] without a sunflower with 3 petals, then $$|\mathcal{F}|\leq3 \left(\begin{array}{c} {n }\\ \lfloor n/3\rfloor \end{array}\right).$$ This result allows us to improve slightly a recent upper bound of Naslund and Sawin for the size of a sunflower-free family in 2[n].