OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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-\varepsilon_0}) strictly separable k-subsets for 1≤ k≤ n/2, with an absolute \varepsilon_0> 0. The constants and positive exponents are nonquantitative.
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MANUSCRIPT 01
A power saving for planar halving lines
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…
Full source 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.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.