Scatter n dots on a page. How many pairs of them can be exactly one centimetre apart?
That is the whole question. Paul Erdos asked it in 1946, and it is one of those problems that sounds like it should have a tidy answer and does not. Call the maximum u(n). If you arrange the dots in a square grid at unit spacing, each interior dot has four neighbours at distance 1, so you get roughly 2n pairs. Linear in the number of points, and not obviously improvable.
Erdos conjectured that you could do a little better but not much: that u(n) grows like n to the power 1 plus something that shrinks away, specifically n^(1+c/log log n). In plain terms, barely more than linear. The belief was that cleverly rescaled grids were essentially optimal.
A gap that stood for forty years
Nobody could prove the conjecture, and nobody could prove an upper bound anywhere near it. The best ceiling, established by Spencer, Szemeredi and Trotter in 1984, is that u(n) is at most a constant times n^(4/3). That is a long way above n^(1+o(1)), and the gap between the two sat open for four decades.
The gap mattered because almost everyone expected the lower end to be right. The grid felt like the natural extremal object, and the exponent 4/3 looked like an artefact of the proof technique rather than the truth.
What happened in May
On 20 May 2026 OpenAI reported that one of its reasoning models had produced a counterexample. Not an improvement to the bound, a disproof of the conjecture: an infinite family of point configurations with at least n^(1+delta) unit distances for some fixed positive delta. That is a polynomial factor above what Erdos predicted, and it rules the conjecture out.
The construction uses algebraic number theory rather than anything combinatorial, which is part of why it had not been found. The same day, Noga Alon, Thomas Bloom, W. T. Gowers, Daniel Litt and Will Sawin posted a short paper giving a digested, human-verified version of the argument. Sawin then pushed the exponent to n^1.014, and it was refined further to n^1.0318. The method appears to run out at roughly n^1.2143, still short of the 4/3 ceiling.
So the headline is accurate but the shape of the contribution is worth being precise about. The model found the construction. Human mathematicians checked it, simplified it, and then improved on it within days. Gil Kalai compared the moment to the 1976 computer-assisted proof of the four colour theorem, which is a reasonable yardstick: a result nobody disputes, arrived at by a route nobody expected.
Why the grid is weak
You can see the weakness of the obvious construction directly. In a k by k lattice, the number of pairs at squared distance m depends entirely on how many ways m splits into two squares, because each such split is a direction the pair can sit in. A squared distance of 1 has four direction vectors. A squared distance of 25 has twelve, because 25 is both 0 plus 25 and 9 plus 16.
In a 20 by 20 grid that difference is 760 pairs against 1,688, from the same 400 points. The equal distances in a grid calculator does this count for any grid and any squared distance, and it shows the other half of the story too: hold the distance at 1 and grow the grid, and the pairs-per-point figure creeps towards 4 and stops. The plain grid is linear, and linear was never going to be the answer.
What it does not mean
It does not mean the unit distance problem is solved. The true growth rate of u(n) is still unknown, and the gap between n^1.0318 and n^(4/3) is wide. What fell was a specific prediction about where in that range the answer sits, and the prediction was wrong in the direction almost nobody expected.
For the formal-proof side of the same year, see the piece on AlphaProof Nexus, and for an earlier machine-found construction see AlphaEvolve and Strassen.