The cap set problem: Up to dimension 7

Abstract

An s-cap n-flat is given by a set of s points, no three of which are on a common line, in an n-dimensional affine space over the field of three elements. The cap set problem in dimension n is: what is the maximum s such that there is an s-cap n-flat? The first two papers in this series of articles considered the cap set problem in dimensions up to and including 5. In this paper, which is the third in the series, we consider dimensions 6 and 7: we prove that every 110-cap 6-flat is a 112-cap 6-flat minus two cap points, and that there are no 289-cap 7-flats.