OpenAI · 2026-05-20 · seismic
OpenAI Model Disproves Erdős's 1946 Unit-Distance Conjecture — First Time AI Has Autonomously Solved a Prominent Open Problem Central to a Field of Mathematics
An internal general-purpose OpenAI reasoning model produced a counter-construction showing the maximum number of unit distances among n points in the plane exceeds n^(1+c) for some fixed c>0 — overturning the long-held belief that grid-like layouts were optimal.
OpenAI says an internal reasoning model autonomously disproved Paul Erdős's 1946 unit-distance conjecture, finding a new construction that beats the previously-believed best bound.
Key specs
| Age of conjecture | 80 years (posed 1946) |
|---|---|
| Result | v(n) >= n^(1+c) for some fixed c>0, for infinitely many n |
| External reviewers | Noga Alon, Melanie Wood, Thomas Bloom |
| Model type | general-purpose reasoning (not math-specialized) |
| Hn traction (at sweep) | 449 points, 288 comments, 2h |
What is it?
OpenAI announced on May 20, 2026 that one of its in-house general-purpose reasoning models — not a system trained or scaffolded specifically for mathematics — found a counter-example to a conjecture Paul Erdős posed in 1946 about the unit-distance problem in discrete geometry. The unit-distance problem asks how many pairs of n points in the plane can be exactly distance 1 apart; Erdős conjectured the maximum, v(n), grows like n^(1+o(1)) — i.e. essentially linearly. OpenAI's model produced a construction showing v(n) >= n^(1+c) for some fixed positive constant c, for infinitely many values of n. The proof was reviewed by three external mathematicians — Noga Alon (Princeton/Tel Aviv), Melanie Wood (Harvard), and Thomas Bloom (Manchester) — who co-authored a companion 'Remarks' paper hosted on OpenAI's CDN.
How does it work?
The model wasn't given a chain-of-thought scaffold, a proof search system, or the Erdős problem as a fine-tuning target. OpenAI describes it as a general-purpose reasoning model that, when prompted on open combinatorics problems, produced an entirely new family of point configurations breaking the long-held belief that the optimum looked roughly like a square grid. The construction yields a polynomial improvement over the previous best lower bound, not just a constant-factor tweak. The companion remarks paper by Alon, Wood and Bloom checks the argument and places it in the context of 80 years of progress on the problem.
Why does it matter?
Until now, AI math wins were either narrow (AlphaProof on IMO problems with formal-proof scaffolding) or vulnerable to embarrassment (Kevin Weil's premature claim seven months ago that GPT-5 had solved 10 Erdős problems, later shown to be re-derivations of known results). OpenAI is framing this as the first time AI has autonomously solved a prominent open problem central to a field of mathematics — a categorically different bar than IMO problem-solving or proof-checking. Combined with Tim Gowers's recent ChatGPT-5.5-Pro session and Terence Tao's verification work, it suggests frontier reasoning models are now genuinely capable of producing new research-level mathematics rather than re-stating known proofs.
Who is it for?
Combinatorics researchers, AI-for-math watchers, and skeptics of AI mathematical claims.
Try it
Read the Alon/Wood/Bloom remarks PDF on OpenAI's CDN for the actual construction and what the c>0 result means.