Upper bounds for sunflower-free sets
Eric Naslund, William F. Sawin

TL;DR
This paper applies the polynomial method to establish upper bounds on sunflower-free set sizes in various combinatorial structures, advancing the understanding of the Erdős–Rado sunflower conjecture.
Contribution
It introduces new bounds for sunflower-free sets using polynomial methods, extending previous results to sets in ext{Z}/D ext{Z}^n and connecting to the Erdős–Rado conjecture.
Findings
Bound for sunflower-free families in subsets of ,2,3n is n +o(1)
Size of sunflower-free sets in D^n is at most c_D^n with explicit c_D
Progress towards proving the Erd53s-Rado sunflower conjecture
Abstract
A collection of sets is said to form a -sunflower, or -system, if the intersection of any two sets from the collection is the same, and we call a family of sets sunflower-free if it contains no sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and Croot, Lev and Pach we apply the polynomial method directly to Erd\H{o}s-Szemer\'{e}di sunflower problem and prove that any sunflower-free family of subsets of has size at most \[ |\mathcal{F}|\leq3n\sum_{k\leq n/3}\binom{n}{k}\leq\left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}. \] We say that a set for is sunflower-free if every distinct triple there exists a coordinate where exactly two of are equal. Using a version of the polynomial method with characters…
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