# Almost-equidistant sets

## TL;DR

This paper investigates the maximum size of almost-equidistant point sets in various dimensions, providing new bounds, constructions, and computer-assisted proofs to advance understanding of their properties.

## Contribution

It offers the first independent, computer-assisted proofs for known extremal values and establishes new bounds and constructions for almost-equidistant sets across multiple dimensions.

## Key findings

- Exact values for f(2) and f(3) confirmed with computer proofs.
- New bounds for f(4), f(5), f(6), f(7), and f(8)–f(9).
- Constructed large almost-equidistant sets in higher dimensions.

## Abstract

For a positive integer $d$, a set of points in $d$-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let $f(d)$ denote the largest size of an almost-equidistant set in $d$-space. It is known that $f(2)=7$, $f(3)=10$, and that the extremal almost-equidistant sets are unique. We give independent, computer-assisted proofs of these statements. It is also known that $f(5) \ge 16$. We further show that $12\leq f(4)\leq 13$, $f(5)\leq 20$, $18\leq f(6)\leq 26$, $20\leq f(7)\leq 34$, and $f(9)\geq f(8)\geq 24$. Up to dimension $7$, our work is based on various computer searches, and in dimensions $6$ to $9$, we give constructions based on the known construction for $d=5$. For every dimension $d \ge 3$, we give an example of an almost-equidistant set of $2d+4$ points in the $d$-space and we prove the asymptotic upper bound $f(d) \le O(d^{3/2})$.