New Lower Bounds for Cap Sets

Abstract

A cap set is a subset of ${\mathbb{F}}_3^n$ with no solutions to $x+y+z=0$ other than when $x=y=z$. In this paper, we provide a new lower bound on 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 $2.218^n$.

Discrete Analysis journal version, 18 pages