Favourite distances in 3-space

Let S be a set of n points in Euclidean 3-space. Assign to each x ∈ S a distance r(x) > 0, and let er(x,S) denote the number of points in S at distance r(x) from x. Avis, Erdo ̋s and Pach (1988) introduced the extremal quantity f3(n) = max ﰝx∈S er(x, S), where the maximum is taken over all n-point subsets S of 3-space and all assignments r: S → (0,∞) of distances. We show that if the pair (S,r) maximises f3(n) and n is sufficiently large, then, except for at most 2 points, S is contained in a circle C and the axis of symmetry L of C, and r(x) equals the distance from x to C for each x ∈ S ∩ L. This, together with a new construction, implies that f3(n) = n2/4 + 5n/2 + O(1).