The cap set problem: Up to dimension 7
Henry Robert Thackeray (University of Pretoria)
TL;DR
This paper advances the understanding of the cap set problem in dimensions 6 and 7 by establishing bounds on the maximum size of cap sets in these dimensions, extending previous work on lower dimensions.
Contribution
It provides new bounds and structural results for cap sets in dimensions 6 and 7, specifically identifying the maximum sizes and configurations of these sets.
Findings
Every 110-cap 6-flat is a 112-cap 6-flat minus two points.
There are no 289-cap 7-flats.
Extended the known bounds of cap sets to higher dimensions.
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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Mathematical Approximation and Integration · Optimization and Packing Problems