Frederick Tyrrell

New lower bounds for cap sets

A cap set is a subset of $\mathbb{F}_3^n$ with no solutions to $x + y + z = 0$ other than when
$x = y = z$. The cap set problem asks how large a cap set can be, and is an important problem in
additive combinatorics and combinatorial number theory. In this talk, I will introduce the problem, give some background and motivation, and describe how I was able to provide the first progress in 20 years on the lower bound for the size of a maximal cap set. Building on a construction of Edel, we use improved computational methods and new theoretical ideas to show that, for large enough $n$, there is always a cap set in $\mathbb{F}_3^n$ of size at least $2.218^n$. No previous knowledge or experience in additive combinatorics is required!