A conditional transfer for colourings of R^n, with the conditional consequence χ(R^3) ≥ 7
- 1Independent researcher
Abstract
Version 2. Remark (2) of Section 10 now states that, conditionally on Theorem 2.3 of the OpenAI preprint, the identity α₁(R²) = m₁(R²) gives the asymptotic statement f(n)/n = m₁(R²) + o(1) of Conjecture 1 of Dúcz and Varga (arXiv:2606.28157); the further statement of that conjecture about Croft's constant is not addressed. The note thanks Dániel Varga for pointing out this connection, and credits Ákos Dúcz for the observation that, conditionally, every Fourier-analytic upper bound on m₁(R²) is approached by the independence ratios of finite unit-distance graphs, with no control on their size. Koshelev, Kupavskii and Voronov (arXiv:2610.18907) independently proved, from the same theorem of OpenAI, that χ(R³) = χm(R³) and hence 7 ≤ χ(R³) ≤ 15; version 2 cites their work in the abstract, the related-work paragraph and Section 9. The mathematics of version 1 is unchanged.
Version 2. Remark (2) of Section 10 now states that, conditionally on Theorem 2.3 of the OpenAI preprint, the identity α₁(R²) = m₁(R²) gives the asymptotic statement f(n)/n = m₁(R²) + o(1) of Conjecture 1 of Dúcz and Varga (arXiv:2606.28157); the further statement of that conjecture about Croft's constant is not addressed. The note thanks Dániel Varga for pointing out this connection, and credits Ákos Dúcz for the observation that, conditionally, every Fourier-analytic upper bound on m₁(R²) is approached by the independence ratios of finite unit-distance graphs, with no control on their size. Koshelev, Kupavskii and Voronov (arXiv:2610.18907) independently proved, from the same theorem of OpenAI, that χ(R³) = χm(R³) and hence 7 ≤ χ(R³) ≤ 15; version 2 cites their work in the abstract, the related-work paragraph and Section 9. The mathematics of version 1 is unchanged.
Assume Theorem 2.3 (rigidity of wild character laws) of the OpenAI preprint The Euclidean plane is not five-colorable. We prove that, for every n ≥ 2 and every set Λ of positive real algebraic numbers, a Λ-avoiding k-colouring of Rn (with no regularity assumption) yields a conull Borel partition of Rn into k sets, each containing no two points at a distance in Λ. Consequently, conditionally on that theorem, χ(Rn) ≥ ⌈1/m₁(Rn)⌉, where m₁(Rn) is the supremum of the upper densities of measurable sets avoiding distance 1. With the published bound on m₁(R³) (DeCorte, de Oliveira Filho and Vallentin) this gives χ(R³) ≥ 7 (previously known lower bound: 6, Nechushtan 2002).
We also derive conditional consequences for unbounded and m-element sets of forbidden distances and for finitary fractional chromatic parameters, whose conditional novelty lies in dimensions 2 and 3. For n ≥ 4 and distances with rational squares, James Davies (arXiv:2610.12301) has independently proved, unconditionally and by a different method, that the chromatic and measurable chromatic numbers coincide; his theorems do not cover dimensions 2 and 3. The transfer in the plane is due to OpenAI; our contribution is its extension to higher dimensions (for n = 3 see also Koshelev, Kupavskii and Voronov, below).
Status. All main results are conditional on Theorem 2.3 of OpenAI's preprint (github.com/openai/math, item 158), which has not been refereed. The note was written with AI assistance (Claude, Anthropic), directed by the author, who is not a professional mathematician, and was checked by independent AI referee passes (OpenAI's ChatGPT and Google's Gemini). It has not yet been reviewed by a human expert. Corrections are very welcome.
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BibTeX
@misc{crocker2026conditional,
title = {{A conditional transfer for colourings of R^n, with the conditional consequence χ(R^3) ≥ 7}},
author = {Carson Crocker},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23633577},
url = {https://zenodo.org/records/23633577},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A conditional transfer for colourings of R^n, with the conditional consequence χ(R^3) ≥ 7
AU - Carson Crocker
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23633577
UR - https://zenodo.org/records/23633577
AB - Version 2. Remark (2) of Section 10 now states that, conditionally on Theorem 2.3 of the OpenAI preprint, the identity α₁(R²) = m₁(R²) gives the asymptotic statement f(n)/n = m₁(R²) + o(1) of Conjecture 1 of Dúcz and Varga (arXiv:2606.28157); the further statement of that conjecture about Croft's constant is not addressed. The note thanks Dániel Varga for pointing out this connection, and credits Ákos Dúcz for the observation that, conditionally, every Fourier-analytic upper bound on m₁(R²) is approached by the independence ratios of finite unit-distance graphs, with no control on their size. Koshelev, Kupavskii and Voronov (arXiv:2610.18907) independently proved, from the same theorem of OpenAI, that χ(R³) = χm(R³) and hence 7 ≤ χ(R³) ≤ 15; version 2 cites their work in the abstract, the related-work paragraph and Section 9. The mathematics of version 1 is unchanged. Version 2. Remark (2) of Section 10 now states that, conditionally on Theorem 2.3 of the OpenAI preprint, the identity α₁(R²) = m₁(R²) gives the asymptotic statement f(n)/n = m₁(R²) + o(1) of Conjecture 1 of Dúcz and Varga (arXiv:2606.28157); the further statement of that conjecture about Croft's constant is not addressed. The note thanks Dániel Varga for pointing out this connection, and credits Ákos Dúcz for the observation that, conditionally, every Fourier-analytic upper bound on m₁(R²) is approached by the independence ratios of finite unit-distance graphs, with no control on their size. Koshelev, Kupavskii and Voronov (arXiv:2610.18907) independently proved, from the same theorem of OpenAI, that χ(R³) = χm(R³) and hence 7 ≤ χ(R³) ≤ 15; version 2 cites their work in the abstract, the related-work paragraph and Section 9. The mathematics of version 1 is unchanged. Assume Theorem 2.3 (rigidity of wild character laws) of the OpenAI preprint The Euclidean plane is not five-colorable. We prove that, for every n ≥ 2 and every set Λ of positive real algebraic numbers, a Λ-avoiding k-colouring of Rn (with no regularity assumption) yields a conull Borel partition of Rn into k sets, each containing no two points at a distance in Λ. Consequently, conditionally on that theorem, χ(Rn) ≥ ⌈1/m₁(Rn)⌉, where m₁(Rn) is the supremum of the upper densities of measurable sets avoiding distance 1. With the published bound on m₁(R³) (DeCorte, de Oliveira Filho and Vallentin) this gives χ(R³) ≥ 7 (previously known lower bound: 6, Nechushtan 2002). We also derive conditional consequences for unbounded and m-element sets of forbidden distances and for finitary fractional chromatic parameters, whose conditional novelty lies in dimensions 2 and 3. For n ≥ 4 and distances with rational squares, James Davies (arXiv:2610.12301) has independently proved, unconditionally and by a different method, that the chromatic and measurable chromatic numbers coincide; his theorems do not cover dimensions 2 and 3. The transfer in the plane is due to OpenAI; our contribution is its extension to higher dimensions (for n = 3 see also Koshelev, Kupavskii and Voronov, below). Status. All main results are conditional on Theorem 2.3 of OpenAI's preprint (github.com/openai/math, item 158), which has not been refereed. The note was written with AI assistance (Claude, Anthropic), directed by the author, who is not a professional mathematician, and was checked by independent AI referee passes (OpenAI's ChatGPT and Google's Gemini). It has not yet been reviewed by a human expert. Corrections are very welcome.
ER -