On almost-equidistant sets

Abstract

A finite set of points in ${\mathbb{R}}^d$ is called almost-equidistant if among any three distinct points in the set, some two are at unit distance. We prove that an almost-equidistant set in ${\mathbb{R}}^d$ has cardinality at most $5d^{13/9}$.

arXiv commentv3: 8 pages, minor corrections. Key words: equilateral sets, almost-equidistant sets, unit distance graph