A Polynomial Improvement of Naslund--Sawin Bound for Sunflower-Free Families Using Triangular Tensors

Abstract

Naslund and Sawin used the slice-rank method for diagonal tensors to prove that $$|\mathcal{F}|=O\!\left(n^{1/2}\left(\frac{3}{2^{2/3}}\right)^n\right)$$ for any sunflower-free family $\mathcal{F}\subseteq 2^{[n]}$. We prove a lemma similar to the slice-rank lemma for the newly defined $i$-triangular tensors, and use it to achieve a polynomial-factor improvement of the bound of Naslund and Sawin by proving that $$|\mathcal{F}|=O\!\left(n^{1/6}\left(\frac{3}{2^{2/3}}\right)^n\right)$$ for any sunflower-free family $\mathcal{F}\subseteq 2^{[n]}$.