Peter Gostev on Hadwiger–Nelson: "five-colours is no go", after "300 billion tokens" of his own attempts

Posted 22:46 UTC on 6 Oct, quoting OpenAI, with a screenshot of the abstract of "The Euclidean plane is not five-colorable" (OpenAI, dated 23 Sep 2026): "I probably spent 300 billion tokens on Hadwiger–Nelson problem (GPT-5.6 and Astra) and got absolutely nothing - and here it is, five-colours is no go, lower bound has moved to 6 or 7. Not a full solution yet, but it's a biggie." The 300-billion-token figure is his own and unverified.

What the repo claims (family 158, checked 7 Oct): every five-colouring of the plane has a monochromatic pair of points at distance one, "with no regularity assumption on the color classes". So the chromatic number of the plane is 6 or 7, and the lower bound rises from 5 (Aubrey de Grey, 2018, https://arxiv.org/abs/1804.02385) to 6; 7 is the classical upper bound. Lean: the scope note (lean/docs/158.md) links comparator statements for "No proper five-coloring" (EuclideanFiveColor.lean: no_proper_five_coloring : ¬ ∃ coloring : ℂ → Fin 5, ProperColoring 5 coloring, for arbitrary colourings) and "Proper seven-coloring" (PlaneColoring.lean). The catalogue's review status is "unchecked". The proofatlas-based list spread by AISafetyMemes marks the problem (#235) "PARTIAL". Reach at 11:31 UTC on 7 Oct: ~59k views, 1,000 likes.

I probably spent 300 billion tokens on Hadwiger–Nelson problem (GPT-5.6 and Astra) and got absolutely nothing - and here it is, five-colours is no go, lower bound has moved to 6 or 7. Not a full solution yet, but it's a biggie.

Quoting @OpenAI: We’re releasing a broad range of new mathematical results produced by an internal frontier model.

We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results.

https://github.com/openai/math

Media: https://pbs.twimg.com/media/HT-1rRGWcAAVnLD.jpg?name=orig

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Archived 2026-10-07 via manual (api.fxtwitter.com, fetched 11:31 UTC).