Rutgers Discrete Mathematics Seminar

Title: Capsets, Sunflower-Free Sets in {0,1}^n, and The Slice Rank Method

Speaker: Eric Naslund, Princeton University

Date: Monday, September 26, 2016 2:00 pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ


A collection of $k$ sets is said to form a $k$-sunflower, or $Delta$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets $mathcal{F}$ sunflower-free if it contains no sunflowers. In this talk we will look at the recent breakthrough of Ellenberg and Gijswijt and Croot, Lev and Pach, which used polynomial method to obtain exponential upper bounds for the Capset problem, that is upper bounds for the size of the largest set in $mathbb{F}_3^n$ which contains no three term arithmetic progressions. In particular we will look at Tao's reformulation of this approach using the so called "Slice Rank Method," and apply it directly to the ErdH{o}s-Szemer'{e}di sunflower problem, proving an exponential upper bound for the size of any sunflower-free family of subsets of ${1,2,…,n}$.