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

TL;DR

This paper introduces a new upper bound for sunflower-free families in combinatorics by combining Tao's slice-rank method with Gr"obner basis techniques, advancing understanding of the Erd ext{"o}s-Rado Sunflower Conjecture.

Contribution

It presents a novel approach that merges algebraic and combinatorial methods to improve bounds on sunflower-free families, specifically for 3-petal sunflowers.

Findings

Proves that a k-uniform sunflower-free family has size at most 3 times a binomial coefficient for certain n and k.

Provides new upper bounds for sunflower-free families in the power set of [n].

Advances the theoretical understanding of the Erd ext{"o}s-Rado Sunflower Conjecture.

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 $2^{[n]}$.