Power savings for planar halving lines and k-sets

What OpenAI claims

Improves the planar halving-line bound to O(n^4/3-ε) 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-ε₀) strictly separable k-subsets for 1 ≤ k ≤ n/2, with an absolute ε₀ > 0. The constants and positive exponents are nonquantitative. (Lean)

Paper

1. A power saving for planar halving lines

   25 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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-ε 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.