The cap set problem and standard diagrams

Abstract

In n-dimensional affine space over the field Z/3Z, a cap is given by a set of points no three of which are in a line, and the cap set problem asks for the largest possible size of an arbitrary cap. The solution to the cap set problem is known for n at most 6.In this paper, we define and apply standard diagrams. These pictures interpret a well-known technique for solving the cap set problem in a new way, allowing conclusions to be derived more easily and intuitively than before. We use standard diagrams to find caps in dimensions up to and including 4 systematically. We prove the apparently new result that in dimension 4, up to isomorphism there are exactly 20 size-18 caps, which we give explicitly.This article is the first of a series. In later articles, we plan to use the methods and results of this paper to investigate dimensions 5 and higher. The eventual goal is to solve the cap set problem in dimension 7, the first unsolved case.