Georgia Institute of Technology College of Sciences

A subset of $\mathbb{F}_3^n$ is called a \emph{cap set} if it does not contain three vectors that sum to zero. In dimension four, this relates to the famous card game SET: a cap set corresponds to a collection of cards without a SET. The cap set problem is concerned with upper bounds on the size of cap sets. The central question raised by Frankl, Graham and R\”odl is: do cap sets have exponentially small density? May last year, this question was (unexpectedly) resolved in a pair of papers by Croot, Lev, and Pach and by Ellenberg and myself in a new application of the polynomial method. The proof is surprisingly short and simple.