{"id":"palomar/PALOMAR-2026-11-03-000009","url":"https://github.com/ukniusanqian-alt/erdos654-lean/tree/3146ec1b4c120e366c10bbb92dec8de596cd7aa1","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-03-000009-v1.json","repo":"https://github.com/ukniusanqian-alt/erdos654-lean","tags":["math.CO","math.MG","msc:52C10"],"score":0,"title":"Erdős Problem 654: the Aletheia counterexample","trust":"high","domain":"math.CO","source":"palomar","authors":["ukniusanqian-alt"],"reviews":0,"subject":"math.CO","summary":"Lean 4 / Mathlib proof that Erdős Problem 654 has a negative answer: for every m ≥ 10 there is a set of n = 4m points in the Euclidean plane, no four on a circle, in which every point has fewer than 3n/4 distinct distances to the other points. Consequently it is false that every n-point planar set with no four concyclic points has a point with (1 - o(1))n distinct distances. The construction (Aletheia; Feng et al., arXiv:2601.22401) places the points (0, ±3^(k+1)) and (±2^(j+1), 0) on the two coordinate axes.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-03","reactions":{},"formalized":true,"formalizes":[{"title":"Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems","identifier":"arXiv:2601.22401","relationship":"formalizes"}]}