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

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

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Eric Naslund , Princeton University
Fine Hall 224

We call a family of sets F "sunflower-free'' if for every three of its sets, some element belongs to exactly two of them. 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 GF(3)^n which contains no three term arithmetic progression. 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 Erdos-Szemeredi sunflower problem, proving an exponential upper bound for the size of any sunflower-free family of subsets of {1,2,…,n}.