Papers › Crescent configurations in normed spaces
Crescent configurations in normed spaces
Sara Fish, Dylan King, Steven J. Miller, Eyvindur A. Palsson, Catherine Wahlenmayer
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We study the problem of crescent configurations, posed by Erd\H{o}s 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 ≤i ≤n - 1, there exists a distance which occurs exactly i times. Constructions of sizes n ≤8 have been provided by Liu, Pal\'{a}sti, and Pomerance. Erd\H{o}s conjectured that there exists some N for which there do not exist crescent configurations of size n for all n ≥N. We extend the problem of crescent configurations to general normed spaces (ℝ², ·) by studying strong crescent configurations in ·. In an arbitrary norm ·, 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 ≤6, n ≤8, and n ≤8 respectively. When defining strong crescent configurations, we introduce the notion of line-like configurations in ·. A line-like configuration in · 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 line-like configurations of arbitrary size. Our main result is a crescent-type result about line-like configurations in the Chebyshev norm. A line-like crescent configuration is a line-like configuration for which no three points lie on a common line and no four points lie on a common · circle. We prove that for n ≥7, every line-like 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.
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