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 $2.218^n$.

翻译：帽集是$\mathbb{F}_3^n$的一个子集，其中除$x=y=z$外不存在满足$x+y+z=0$的解。本文基于Edel的构造，通过改进的计算方法和新的理论思路，给出了最大帽集大小的新下界。我们证明，对于足够大的$n$，$\mathbb{F}_3^n$中始终存在大小至少为$2.218^n$的帽集。