Erdős Problem 657: Straus's observation on isosceles-free point sets with few distances
Erdős asked whether a set of n points in the plane with no isosceles triangle must determine at least f(n)n distinct distances for some f(n) tending to infinity, and asked the same in R^k for each fixed k. Straus observed that no such bound holds uniformly in the dimension. In the form stated in formal-conjectures: for every k there are 2^k points in R^k with no isosceles triangle and at most 2^k - 1 distinct distances. This formalization proves that statement, using the points whose i-th coordinate is 0 or sqrt(2^i); the sources consulted do not record Straus's own construction.
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