Crescent Configurations

In 1989, Erd˝os conjectured that for a suciently large n it is impossible to place n points in general position in a plane such that for every 1 ≤ i ≤ 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 ≤ 8. The one for n = 5 was due to Pomerance while Pal´asti 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 suciently 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 suciently large dimension d such that there is a configuration in d-dimensional space meeting Erd˝os’ criteria.

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