New Lower Bounds for Cap Sets

TL;DR

This paper establishes a new lower bound on the size of maximal cap sets in vector spaces over finite fields, demonstrating that large cap sets of size at least 2.218^n exist for sufficiently large dimensions.

Contribution

It introduces improved computational techniques and theoretical insights to prove the existence of larger cap sets than previously known.

Findings

Existence of cap sets of size at least 2.218^n for large n

Enhanced methods for constructing large cap sets

Advances in theoretical understanding of cap set bounds

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 $2.218^n$.