Classification of crescent configurations

TL;DR

This paper classifies crescent configurations of 4 and 5 points using graph isomorphism and rigidity, offering new insights and methods that could extend to higher dimensions in the study of these geometric arrangements.

Contribution

It provides a systematic classification of small crescent configurations via graph isomorphism and rigidity, introducing techniques that can be generalized to higher dimensions.

Findings

Classified crescent configurations for 4 and 5 points.

Developed a new approach using graph isomorphism and rigidity.

Proposed methods extend to higher-dimensional configurations.

Abstract

Let $n$ points be in crescent configurations in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$ times. Since Erd\H{o}s' conjecture in 1989 on the existence of $N$ sufficiently large such that no crescent configurations exist on $N$ or more points, he, Pomerance, and Pal\'asti have given constructions for $n$ up to $8$ but nothing is yet known for $n \geq 9$. Most recently, Burt et. al. had proven that a crescent configuration on $n$ points exists in $\mathbb{R}^{n-2}$ for $n \geq 3$. In this paper, we study the classification of these configurations on $4$ and $5$ points through graph isomorphism and rigidity. Our techniques, which can be generalized to higher dimensions, offer a new viewpoint on the problem through the lens of distance geometry and provide a systematic way to construct crescent configurations.