{"id":"hexagon/2610.00534","pdf":"https://hexagonmath.org/pdf/2610.00534v1","url":"https://hexagonmath.org/abs/2610.00534v1","tags":["msc:51D20","combinatorial-geometry"],"score":0,"title":"The maximum cardinality of a planar seven-distance set","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Alon Amit"],"reviews":0,"subject":"math.CO","summary":"We give a computer-assisted proof that the maximum cardinality of a planar set determining at most seven distinct positive distances is sixteen. A diameter-graph argument reduces a hypothetical seventeen-point example to an eleven-point set with at most six distances, except for a thirteen-cycle obstruction. We exclude the cycle by an exact interval certificate. Using a published exhaustive count of sixty-two candidate distance patterns, we classify the eleven-point cores by exact Gröbner-basis calculations and prove that none can have the required seventeen- point extension. All extension…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}