The cap set problem: 41-cap 5-flats

TL;DR

This paper classifies all 5-dimensional caps of size at least 41 in the context of the cap set problem over Z/3Z, advancing understanding of maximal cap sizes in affine spaces.

Contribution

It systematically classifies all 5-dimensional caps of size ≥41, extending previous work on the cap set problem in higher dimensions.

Findings

Identified all 5-dimensional caps of size at least 41

Extended classification methods from previous research

Contributed to understanding maximal cap sizes in affine spaces

Abstract

An s-cap n-flat, or an n-dimensional cap of size s, is a pair (S,F) where F is an n-dimensional affine space over Z/3Z and the size-s subset S of F contains no triple of collinear points. The cap set problem in dimension n asks for the largest s for which an s-cap n-flat exists. This series of articles investigates the cap set problem in dimensions up to and including 7. This is the second paper in the series. By applying and adapting methods from the first paper in the series, we systematically classify all 5-dimensional caps of size at least 41.