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 $\mathcal{F}\subseteq 2^{[n]}$.

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