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A Proof of a Dodecahedron Conjecture for Distance Sets

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Abstract

A finite subset of a Euclidean space is called an s-distance set if there exist exactly s values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in \(\mathbb {R}^3\) is 20, and every 5-distance set in \(\mathbb {R}^3\) with 20 points is similar to the vertex set of a regular dodecahedron.

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Acknowledgements

The authors thank Kenta Ozeki for providing information on the papers [11, 12] relating to graphs without two vertex-disjoint odd cycles. Nozaki is supported by JSPS KAKENHI Grant Numbers 19K03445 and 20K03527. Shinohara is supported by JSPS KAKENHI Grant Number 18K03396.

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Correspondence to Masashi Shinohara.

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Nozaki, H., Shinohara, M. A Proof of a Dodecahedron Conjecture for Distance Sets. Graphs and Combinatorics 37, 1585–1603 (2021). https://doi.org/10.1007/s00373-021-02318-5

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  • DOI: https://doi.org/10.1007/s00373-021-02318-5

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