Popular distances in 3-space

Erdős, Paul and Harcos, Gergely and Pach, János (1999) Popular distances in 3-space. Discrete Mathematics, 200 (1-3). pp. 95-99. ISSN 0012-365X

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Abstract

Let m(n) denote the smallest integer m with the property that any set of n points in Euclidean 3-space has an element such that at most m other elements are equidistant from it. We have that cn(1/3) log log n less than or equal to m(n) less than or equal to n(3/5) beta(n), where c>0 is a constant and beta(n) is an extremely slowly growing function, related to the inverse of the Ackermann function.

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