Improves the planar halving-line bound to O(n^{4/3-\varepsilon}) for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has O(n(k+1)^{1/3-\varepsilon_0}) strictly separable k-subsets for 1\le k\le n/2, with an absolute \varepsilon_0\gt 0. The constants and positive exponents are nonquantitative. (Lean)
Manuscript
A power saving for planar halving lines
September 25, 2026 · source · cite as \cite{OAI:A-power-saving-for-planar-halving-lines-September-25-2026}
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.
Result 183 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md