{"id":"openai-math/A-power-saving-for-planar-halving-lines","pdf":"https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026","tags":["discrete-geometry","halving-lines","combinatorics"],"score":0,"title":"A power saving for planar halving lines","domain":"Combinatorics","family":"183","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"There are absolute constants ε > 0 and C such that every sufficiently large even n -point set in the plane with no three collinear has at most $`Cn^{4/3-\\varepsilon}`$ unordered halving pairs. This gives a power saving over the classical $`O(n^{4/3})`$ bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"Power savings for planar halving lines and k -sets"}