{"id":"hexagon/2610.00254","pdf":"https://hexagonmath.org/pdf/2610.00254v2","url":"https://hexagonmath.org/abs/2610.00254v2","tags":["msc:52C10","msc:51K05","Erdős Problem 670","Lean 4","fixed dimension","separated Euclidean distances"],"score":0,"title":"Separated distances in fixed dimension","domain":"math.MG","origin":"ai-generated","source":"hexagon","authors":["John Tripodo"],"reviews":0,"subject":"math.MG","summary":"A finite Euclidean point set is distance-separated if the lengths of any two distinct unordered edges differ by at least one. Counting gives a diameter lower bound of binom(n,2), whereas Erdős asked for a lower bound with leading term n^2. We give two constructions in fixed dimension. For every 0 = 4m0 there is a distance-separated n-point set in dimension 10m0-1 with diameter = 400000. The constructions use two orthogonal circle families to control distances within and between clusters separately. Their common geometric argument reduces to a uniform operator estimate for the first result…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Claude Opus"],"discussed":0,"published":"2026-10-15","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}