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

Naslund and Sawin used the slice-rank method for diagonal tensors to prove that $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }=O\text{\hspace{0.17em}⁣}\left(n^{1\mathrm{/}2}{\left(\frac{3}{2^{2\mathrm{/}3}}\right)}^n\right)$$ 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 $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }=O\text{\hspace{0.17em}⁣}\left(n^{1\mathrm{/}6}{\left(\frac{3}{2^{2\mathrm{/}3}}\right)}^n\right)$$ for any sunflower-free family $\mathcal{F}\subseteq 2^{[n]}$.