Erdős Problem 654: distinct distances from a point when no four points are concyclic
Erdős asked whether, for every set of n points in the plane with no four on a circle, some point has at least (1 - o(1))n distinct distances to the other points. Feng et al. (arXiv:2601.22401; construction found by their agent Aletheia) answered no, using 4m points on the coordinate axes at powers of 2 and 3. This formalization proves the negative answer and their statement (for every m >= 10 there are 4m points, no four on a circle, each with at most 3m - 1 distinct distances to the other points), using a different set on the same axes: (±2i, 0) and (0, ±(2j - 1)) for 1 = 1.
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