In n-dimensional affine space over the field $\mathbb{Z}/3\mathbb{Z}$, 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.

在场上的n维仿射空间中$\mathbb{Z}/3\mathbb{Z}$，一个上限由一组没有三个点在一条线上的点给出，并且上限集问题要求任意上限的最大可能大小。上限集问题的解已知n最多为 6。

在本文中，我们定义并应用标准图表。这些图片以一种新的方式解释了解决上限集问题的众所周知的技术，可以比以前更容易、更直观地得出结论。我们使用标准图表来系统地查找尺寸最大为 4 的大写字母。我们证明了一个明显的新结果，在第 4 维中，直到同构，正好有 20 个大小为 18 的上限，我们明确给出。

这篇文章是系列文章的第一篇。在后面的文章中，我们计划使用本文的方法和结果来研究维度 5 及更高维度。最终目标是解决第 7 维中的上限集问题，这是第一个未解决的情况。