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Discrete Math

Capsets, sunflower-free sets in {0,1}^n, and the slice rank method

Location:  Hill 705
Date & time: Monday, 26 September 2016 at 2:00PM - 2:11PM

Eric Naslund, Princeton: A collection of \(k\) sets is said to form a \(k\)-sunflower, or \(\Delta\)-system, if the intersection of any two sets from thecollection 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 Erd\H{o}s-Szemer'{e}di sunflower problem, proving an exponential upper bound for the size of any sunflower-free family of subsets of \(\{1,2,…,n\}\).'

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Department of Mathematics

Department of Mathematics
Rutgers University
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