Result 183, Combinatorics

Power savings for planar halving lines and k-sets

Improves the planar halving-line bound to O(n4/3−ε)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−ε0)O(n(k+1)^{1/3-\varepsilon_0}) strictly separable k-subsets for 1≤k≤n/21\le k\le n/2, with an absolute ε0>0\varepsilon_0\gt 0. The constants and positive exponents are nonquantitative.

Lean formalization New or sharp bound

The bigger picture

Why it matters

A line through two points is a halving line if it leaves equally many remaining points on either side. The manuscript claims that planar point sets admit fewer such balanced splits than the classical bound allows.

What changes?

For every sufficiently large even n, the reported bound on unordered halving pairs is C times n raised to (4/3 minus epsilon), assuming no three points are collinear. The summary also gives O(n times (k+1) raised to (1/3 minus epsilon_0)) strictly separable k-subsets for n-point planar sets under the same collinearity assumption, with k between 1 and n/2. Both epsilon and epsilon_0 are absolute positive constants. The proof supplies no explicit values for the constants or exponent savings.

What does that help mathematicians do?

A strictly separable k-subset is a group of k points that a line can place entirely on one side, with all other points on the other. The claimed bounds limit how many distinct groups can arise this way. For halving pairs, the improvement rules out counts proportional to n raised to 4/3 for arbitrarily large sets, not merely a particular constant in the classical estimate.

Are there practical applications?

The immediate value is foundational: sharper limits on line-induced partitions refine the combinatorial understanding of planar point configurations. They can constrain counting arguments involving balanced splits or separable subsets. The sources do not establish a faster algorithm, and the nonquantitative constants prevent extracting an explicit numerical improvement for a given data size.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A power saving for planar halving lines

September 25, 2026 31 pages

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 Cn4/3−εCn^{4/3-\varepsilon} unordered halving pairs. This gives a power saving over the classical O(n4/3)O(n^{4/3}) bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-power-saving-for-planar-halving-lines-September-25-2026,
  author = {{OpenAI}},
  title = {{A power saving for planar halving lines}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf}{OAI:A-power-saving-for-planar-halving-lines-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/183.md.

Power savings for planar halving lines and k-sets

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

A halving pair in an even planar point set is a pair whose line leaves equally many remaining points on each side. The formalization proves that some absolute ε>0\varepsilon>0 and CC bound the number of halving pairs by Cn4/3−εCn^{4/3-\varepsilon} for every sufficiently large even nn and every nn-point set with no three collinear. It also proves a bound of the same form for all level-switch counts under the additional generic-position assumptions in the statement. The constants are existential.

Comparator links

Result Comparator statement
Power-saving bounds for halving pairs and level switches HalvingLines.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.