Crescent configurations in normed spaces
Abstract
We study the problem of crescent configurations, posed by Erdős in 1989. A crescent configuration is a set of $n$ points in the plane such that: 1) no three points lie on a common line, 2) no four points lie on a common circle, 3) for each $1 \leq i \leq n  1$, there exists a distance which occurs exactly $i$ times. Constructions of sizes $n \leq 8$ have been provided by Liu, Palásti, and Pomerance. Erdős conjectured that there exists some $N$ for which there do not exist crescent configurations of size $n$ for all $n \geq N$. We extend the problem of crescent configurations to general normed spaces $(\mathbb{R}^2, \ \cdot \)$ by studying strong crescent configurations in $\ \cdot \$. In an arbitrary norm $\\cdot \$, we construct a strong crescent configuration of size 4. We also construct larger strong crescent configurations in the Euclidean, taxicab, and Chebyshev norms, of sizes $n \leq 6$, $n \leq 8$, and $n \leq 8$ respectively. When defining strong crescent configurations, we introduce the notion of linelike configurations in $\\cdot \$. A linelike configuration in $\\cdot \$ is a set of points whose distance graph is isomorphic to the distance graph of equally spaced points on a line. In a broad class of norms, we construct linelike configurations of arbitrary size. Our main result is a crescenttype result about linelike configurations in the Chebyshev norm. A linelike crescent configuration is a linelike configuration for which no three points lie on a common line and no four points lie on a common $\\cdot \$ circle. We prove that for $n \geq 7$, every linelike crescent configuration of size $n$ in the Chebyshev norm must have a rigid structure. Specifically, it must be a perpendicular perturbation of equally spaced points on a horizontal or vertical line.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 DOI:
 10.48550/arXiv.1909.08769
 arXiv:
 arXiv:1909.08769
 Bibcode:
 2019arXiv190908769F
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 27 pages, 15 figures