{"id":"palomar/PALOMAR-2026-11-02-000006","url":"https://github.com/veljjanoski/erdos654-palomar/tree/fd7a8dc3cf52f6dcadcf5c6bcd05a2dc387dc64f","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-02-000006-v1.json","repo":"https://github.com/veljjanoski/erdos654-palomar","tags":["math.CO","math.MG","msc:52C10"],"score":0,"title":"Erdős Problem 654: distinct distances from a point when no four points are concyclic","trust":"high","domain":"math.CO","source":"palomar","authors":["Daniel Veljjanoski"],"reviews":0,"subject":"math.CO","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-02","reactions":{},"formalized":true,"formalizes":[{"title":"Some of my favourite unsolved problems","identifier":"","relationship":"formalizes"},{"title":"Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems","identifier":"arXiv:2601.22401","relationship":"adapts"}]}