SCHWEITZER 1994 Problem 7

Prove that there exist $0 < \alpha< \beta<1$ numbers have the following properties. (i) for any sufficiently large n, n points can be specified in $\Bbb R^3$ , so that each point is equidistant from at least $n^\alpha$ other points. (ii) the above statement is no longer true with $n^\beta$ instead of $n^\alpha$