Crescent configurations

TL;DR

This paper generalizes Erdős's conjecture on point configurations with unique distance counts to higher dimensions, providing constructions that demonstrate such configurations exist in sufficiently large dimensions for any number of points.

Contribution

It introduces a new construction method in higher dimensions proving the existence of configurations meeting Erdős's criteria for any number of points.

Findings

Existence of configurations in high dimensions for any n

Construction method applicable in d-dimensional space

Extension of Erdős's conjecture to higher dimensions

Abstract

In 1989, Erd\H{o}s conjectured that for a sufficiently large $n$ it is impossible to place $n$ points in general position in a plane such that for every $1\le i \le n-1$ there is a distance that occurs exactly $i$ times. For small $n$ this is possible and in his paper he provided constructions for $n\leq 8$. The one for $n=5$ was due to Pomerance while Pal\'{a}sti came up with the constructions for $n=7,8$. Constructions for $n=9$ and above remain undiscovered, and little headway has been made toward a proof that for sufficiently large $n$ no configuration exists. In this paper we consider a natural generalization to higher dimensions and provide a construction which shows that for any given $n$ there exists a sufficiently large dimension $d$ such that there is a configuration in $d$-dimensional space meeting Erd\H{o}s' criteria.