erdős

The problem

In 1946, Paul Erdős asked: place n points in the Euclidean plane. How many pairs of points can be at exactly the same distance?

The obvious construction is a square grid. Erdős showed that an √n × √n grid, with a careful choice of unit, yields roughly n^{1 + c/log log n} pairs at unit distance — the exponent creeps just barely past 1. He conjectured this was essentially optimal: that no arrangement could push the count above n^{1 + o(1)}.

The conjecture stood for 80 years. Erdős attached a $500 prize. Spencer, Szemerédi, and Trotter proved a matching upper bound of n^4/3 in 1984, narrowing the gap from above. The unit-distance conjecture itself resisted.

May 20, 2026

An internal reasoning model at OpenAI produced a proof that the conjecture is false. There exists ε > 0 and an infinite family of point configurations achieving at least n^1+ε unit-distance pairs. Will Sawin refined the explicit constant to ε ≥ 0.014.

That's a polynomial improvement over Erdős's lower bound — qualitatively different from sub-polynomial growth. The conjecture isn't merely imprecise; it's wrong.

The proof was checked and digested in a 19-page paper by Noga Alon, Thomas Bloom, Tim Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood. Bloom — who had torched OpenAI's previous Erdős claim seven months ago — verified this one personally.

The classical construction (1946)

The grid tab is the construction Erdős discovered: place points on a square integer grid. The number of pairs at squared distance m is roughly n · r₂(m) / 2, where r₂(m) is the number of integer solutions to a² + b² = m. When m is a product of k distinct primes ≡ 1 mod 4, r₂(m) grows like 4 · 2^k. Pick m = 5 · 13 · 17 · 29 · 37 · …, push k as far as the grid allows, and you get Erdős's lower bound of n^{1 + c/log log n}.

Algebraically: these grid points live in the ring of Gaussian integers ℤ[i], the ring of integers in the field ℚ(i). This is a CM field — a totally imaginary quadratic extension of a totally real field. Its degree over ℚ is n = 2; the geometry-of-numbers parameter is f = [ℚ(i)⁺:ℚ] = 1.

This is the picture mathematicians carried for 80 years. The criss-crossing in the grid tab is its hero image. It's the before.

The new construction (2026)

Replace ℚ(i) with a CM field K of growing degree. The new construction:

1. Pick a CM field K = L(i) where L is totally real, with f = [L:ℚ] → ∞.
2. Embed the ring of integers O_K as a lattice in ℂ^f via the Minkowski map (one chosen complex embedding per conjugate pair).
3. Find a large set U of magnitude-one elements u ∈ K* — these satisfy |σ(u)| = 1 for every complex embedding σ. They come from prime splitting in K followed by a pigeonhole on the class group of K.
4. Take all lattice points inside a polydisc {|z_r| ≤ R}^f.
5. Project to one complex coordinate. The projected pairs (x, x+u) for u ∈ U are exactly at distance 1 in ℂ ≅ ℝ², because |σ₁(u)| = 1.

To make the lattice well-shaped while [K:ℚ] grows, the proof uses an infinite Golod–Shafarevich class field tower over a cyclic cubic base field F — totally-real fields with bounded root discriminant whose degrees go to infinity, in which a fixed prime q ≡ 1 mod 4 splits completely at every level. Sawin's refinement turns this into the explicit exponent ε ≥ 0.014.

The point set is not a grid. It's the irregular 2D shadow of a 4D, 6D, …, 2fD lattice — the points cluster by algebraic structure, with the polydisc envelope projecting to (something near) a disc rather than a square. Nobody has published a figure of this: the proof papers contain zero diagrams. The cm-field tab is our own faithful realisation at the smallest non-trivial case (f = 2, the degree-4 CM fields ℚ(ζ₅), ℚ(ζ₈), ℚ(ζ₁₂)), enumerated, projected, and decorated with the unit-distance pairs it actually produces. At f = 2 you don't get the polynomial improvement yet — you need f → ∞ — but you can see the shape.

Why mathematicians missed this

Several blockers led people away from this approach:

• Erdős believed the conjecture was true. His prize was payable for either direction, but the community spent decades trying to prove the upper bound.
• The first natural generalization — varying primes in a fixed CM field — gives no asymptotic improvement; the prime number theorem makes the choice of field invisible. You have to vary the field itself.
• Pursuing increasing-degree number fields means working in high-dimensional lattices and projecting to 2D. In Tsimerman's words, "a very scary dynamic" — it consumes attention and frequently doesn't work out.

The AI tried it anyway. One line from its chain of thought, quoted in the remarks paper:

…in principle all extremal examples can be taken algebraic. But the degree and height of that algebraic realization can be enormous… Maybe that enormous degree is not just an annoyance but a source of possible counterexamples. Number fields deserve a closer look.

What's in this site

• grid — the classical 1946 construction. A k×k grid of integer lattice points (Gaussian integers, if you like), with edges between every pair at squared distance m. Pick m with many distinct prime factors ≡ 1 mod 4 — the presets walk through 1, 5, 25, 65, 325, 1105, 32045 — and watch the edge count explode. The eight-direction line bundle visible at m = 5 is what gives the iconic image its octagonal-star feel; that's the classical construction looking its best.
• cm-field — our faithful realisation of the new 2026 construction at f = 2. Pick a degree-4 CM field ℚ(ζ₅), ℚ(ζ₈), or ℚ(ζ₁₂); choose a polydisc radius R; the page enumerates O_K lattice points in the bidisc, projects to one ℂ coordinate, and finds the actual unit-distance pairs in that projection. The shape is qualitatively new: no grid, an irregular cloud with the symmetry of K, edges arising from cross-dimensional algebraic structure rather than from a 2D lattice basis.
• growth — log-log chart of the bounds: Erdős's lower (1946) — slightly super-linear; OpenAI / Sawin's new lower (2026) — n^1.014, polynomial in n; Spencer–Szemerédi–Trotter upper (1984) — n^4/3; and the disproven conjecture region. The new lower bound visibly pokes out of the disproven band.
• docs — this page.

The gap between the new lower bound (n^1.014) and the SST upper bound (n^1.333) is now polynomially open. The next chapter of the story.

sources

• OpenAI announcement, May 20 2026
• Remarks on the disproof of the unit distance conjecture — Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, Wood (19 pp)
• An explicit lower bound for the unit distance problem — Sawin (the ε = 0.014 refinement)
• Amazing: Erdős's Unit Distance Problem was Disproved! — Gil Kalai's blog
• erdosproblems.com #90 — the curated catalog entry
• Erdős, P. (1946). On sets of distances of n points. American Mathematical Monthly 53, 248–250 — the original.