CRESCENT CONFIGURATIONS

摘要

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.