OpenAI raises the plane's chromatic number lower bound to 6, Lean-formalized
fortnow · x · 2026-10-12
Blogger Bill Gasarch reviews OpenAI's claimed solutions to a batch of math/TCS conjectures, with Lance Fortnow having collected the TCS-related results on a website.
- Key case: the chromatic number of the plane. For nearly 70 years only 4≤χ≤7 was known; in 2018 Aubrey de Grey raised the lower bound to 5 with a 1,581-vertex graph, later reduced to 509 vertices by Jaan Parts.
- OpenAI's new result shows 6≤χ, formalized in Lean.
- Notably, the proof did not simply search for larger graphs. Instead it took a different approach: proving that if a proper 5-coloring of R² exists, then a weak measurable proper 5-coloring exists — and that no such weak measurable proper 5-coloring exists.
- The author deliberately brackets questions about what this means for the future of mathematics and academia.
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