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Measurable Independence Density Equals the Finite Independence-Ratio Infimum in the Euclidean Plane

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.MG — Metric Geometry

Contributed by Ákos Dúcz ↗

Version 1 / Oct 09, 2026 / CC BY 4.0

Abstract

Let m1(R^2) be the supremum of upper densities of Lebesgue-measurable subsets of the plane containing no pair of points at distance one. We show, using the spectral rigidity theorem in OpenAI's recent proof that the plane is not five-colorable, that m1(R^2) = inf_G alpha(G)/|V(G)|, where G ranges over nonempty finite unit-distance graphs in the plane. The argument constructs an isometry-invariant law on independent subsets of the countable algebraic plane, projects the occupancy indicator onto the continuous spectral factor without changing its expectation, and extracts a measurable independent set with arbitrarily small density loss. As a consequence, if f(n) is the least independence number among unit-distance graphs on n vertices, then f(n)/n converges to m1(R^2). The proof is nonquantitative and uses the cited spectral rigidity result as a black box.

Provenance statement

The results relies on the new techniques introduced by OpenAI's "The Euclidean plane is not five colorable" paper. ChatGPT 6 was asked produce this result using the new techniques from that paper, in particular using the weak-measurable coloring reduction. Apart from this minimal amount of guideance, the work was done entirely by ChatGPT.

References

  1. G. Ambrus, A. Csiszárik, M. Matolcsi, D. Varga, and P. Zsámboki, The density of planar sets avoiding unit distances , Math. Programming (2023), https://doi.org/10.1007/s10107-023-02012-9 doi:10.1007/s10107-023-02012-9 ; https://arxiv.org/abs/2207.14179 arXiv:2207.14179 .arXivDOI
  2. Á. Dúcz and D. Varga, A unit-distance graph in the plane with independence ratio below 1/41/4 , preprint (2026), https://arxiv.org/abs/2606.28157 arXiv:2606.28157 .arXiv
  3. M. Matolcsi, I. Z. Ruzsa, D. Varga, and P. Zsámboki, The fractional chromatic number of the plane is at least 44 , preprint (2023; revised 2025), https://arxiv.org/abs/2311.10069 arXiv:2311.10069 .arXiv
  4. OpenAI, The Euclidean plane is not five-colorable , OpenAI Math Release preprint, September 23, 2026, https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf online manuscript .link

Version history

  1. v1Submitted by Ákos DúczInitial depositCurrentOct 09, 2026