Mathematics · Discrete Geometry · Number Theory

Planar Unit Distance ProblemErdős's 1946 conjecture — autonomously disproved by AI, 80 years later

OpenAI Blog PDFreleased 2026-05-20

Given $n$ points in the plane, how many pairs can be at distance exactly $1$ ? That's Erdős's 1946 unit-distance problem — one of the most famous open questions in discrete geometry. He conjectured $ν (n) \leq n^{1 + C / l o g l o g n}$ , a subpolynomial growth. For 40 years the best upper bound was Spencer–Szemerédi–Trotter's $O (n^{4/3})$ ; for 80 years the best lower bound was Erdős's own grid construction. In 2026, OpenAI's internal model, working fully autonomously, found a counterexample: there is a constant $δ > 0$ such that $ν (n) \geq n^{1 + δ}$ holds for infinitely many $n$ — the conjecture is false.

This page walks through it with 5 interactive components: from the bare problem statement to the heart of the construction (a Golod–Shafarevich tower, CM fields, Minkowski embedding).

1946 — Erdős conjecture

n^{1+C/log log n}

upper bound — now disproved

Am. Math. Monthly 53

1984 — SST upper

O(n^4/3)

best upper bound for 40 years

Graph Theory & Comb.

2026 — new lower

n^1+δ

counterexample — Erdős conjecture refuted

OpenAI 2026

AI meta-noteThe paper has a "Statement on AI Use" section. The problem statement was AI-authored, the internal model solved it fully autonomously, an AI grading pipeline scored the first solution as correct, and only then were human researchers brought in. Number-theory experts later verified, simplified, and strengthened the argument; the manuscript is a human-edited exposition of the AI-original output.

TL;DR

What: let $ν (n)$ = max number of unit-distance pairs among n planar points. Erdős 1946 conjectured $ν (n) \leq n^{1 + C / l o g l o g n}$ .
Counter: there is $δ > 0$ such that $ν (n) \geq n^{1 + δ}$ for infinitely many n.
How: lift Erdős's classical ℤ[i] grid trick to high dimensions. Use an infinite unramified pro-3 tower over a totally-real field L (degrees → ∞, root discriminant bounded), take the CM field K = L(i); Chebotarev gives primes that split completely in every layer, producing exponentially many u with |σ(u)| = 1. Minkowski-embed into ℂ^f and project to the first coordinate to get an ℝ² point set.
Key tools: Golod–Shafarevich keeps the pro-3 tower infinite; a Frattini-subgroup argument adds splitting conditions without dropping the generator rank; CM + Minkowski translates "algebraic norm 1" to "geometric |σ(u)| = 1".
AI angle: AI wrote the problem, the internal model solved it autonomously, an AI grading pipeline scored it, then humans reviewed. The raw model output is published alongside the paper.

3. The problem — drag the points

Given n distinct points $P \subset R^{2}$ in the plane, define

ν (P) = # {{x, y} \subset P : ∣ x - y ∣ = 1}, ν (n) = ∣ P ∣ = n max ν (P) .

Drag any point below. Equilateral triangles maximise local ν/n (every triangle edge is a unit distance); switching to the square lattice loses the diagonal family entirely.

3.5 The upper bound — two circles meet in ≤ 2 points

The other direction: why can't ν(n) be too large? Simplest fact: the unit-distance graph contains no $K_{2, 3}$ . If points x, y are both at distance 1 from z₁, z₂, z₃, then z₁, z₂, z₃ lie on the unit circle around x and on the unit circle around y. Two distinct circles meet in at most 2 points, so at most 2 such z's exist — not 3. Hence the unit-distance graph is K₂,₃-free.

By Kővári–Sós–Turán (1954), any K₂,₃-free graph has ≤ $\frac{1}{2} (1 + 4 n - 3) n^{1/2} \cdot n = O (n^{3/2})$ edges. Erdős's 1946 paper used exactly this to bound ν.

In 1984 Spencer–Szemerédi–Trotter sharpened this to $O (n^{4/3})$ via their point-curve incidence theorem (proved that same year). In 1997 Székely gave a one-page crossing-number proof: draw the unit-distance graph in the plane, every unit distance is an arc on some unit circle; the Crossing Lemma $cr (G) \geq c \cdot e^{3} / n^{2}$ directly forces $e = O (n^{4/3})$ . This is the best upper bound to date — only the constant has been improved (Ágoston–Pálvölgyi 2022).

4. Erdős's 1946 trick — redefine "unit"

↑ Full-resolution exact reproduction — same construction as OpenAI's hero figure. Every segment is exactly √k long. Switch k to feel how r₂(k) drives density.

Take the √n × √n integer grid. With "unit = 1", there are only $2 s (s - 1) \approx 2 n$ unit-distance pairs — a trivial linear bound. Erdős's key observation: redefine "unit distance" to be $k$ for any integer k. Now two points (x₁, y₁), (x₂, y₂) are at unit distance ⟺ $(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} = k$ ⟺ k is a sum of two squares.

Let $r_{2} (k)$ count the ways to write k as a sum of two squares. r₂(1) = 4 ((±1, 0), (0, ±1)); r₂(5) = 8 ((±1, ±2), (±2, ±1)). In the s×s grid, the count of pairs at squared distance k is ≈ $\frac{1}{2} \sum_{(a, b) : a^{2} + b^{2} = k} (s - ∣ a ∣) (s - ∣ b ∣)$ , proportional to r₂(k).

Choosing k ≤ n with maximum r₂(k), Erdős got

ν (n) \geq n \cdot n^{c / l o g l o g n} = n^{1 + c / l o g l o g n} .

The same form as his conjectured upper! He believed his own lower bound was tight up to constants. Below, slide k and s to see how r₂(k) drives ν:

4.5 Why Erdős (and everyone else) believed the conjecture

Erdős's grid gave 1 + c/log log n, which approaches 1 ("almost linear"); his upper bound O(n^(3/2)) was improved by SST to O(n^(4/3)), still leaving room for n^(1+o(1)). Decades of subsequent work all pointed toward n^(1+o(1)).

2011

Matoušek

For most (Baire-generic) norms on ℝ², ν ≤ $O (n lo g n lo g lo g n)$ — near-linear.

2025

Alon–Bucić–Sauermann

For comeagre norms on ℝ^d, ν ≤ $\frac{d}{2} n lo g^{2} n$ — near-linear in all dims.

2025

Greilhuber–Schildkraut–Tidor

For all norms (not just comeagre), matching lower bound $(d /2 - o (1)) n lo g^{2} n$ . Even stronger evidence.

Why Euclidean breaks the patternAll "generic-norm near-linear" results depend on the unit ball lacking arithmetic structure. But the Euclidean circle

x^{2} + y^{2} = 1

has an extremely special arithmetic structure: in Gauss integers ℤ[i], "norm = 1" is

z \overset{z}{ˉ} = 1

, the entry point of number theory. Lifted to CM fields K = L(i), elements with |σ(u)| = 1 in every complex embedding come from the same algebraic machinery. Erdős wasn't wrong because the intuition "should be near-linear" was bad — he was wrong because he never considered that Euclidean is itself the counterexample. That's what the AI noticed.

Bloom in Remarks §4 puts it bluntly: "Most of the human effort has been on trying to prove the upper bound, rather than disprove it." — 80 years of effort in the wrong direction.

5. 80 years of bound-tightening

1946

n·c√(log n / log log n)≤ ν ≤O(n^3/2)Erdős

Erdős states the problem + conjecture ν(n) ≤ n^(1+C/log log n); K₂,₃-free gives upper O(n^3/2)

1984

n·c√(log n / log log n)≤ ν ≤O(n^4/3)SST

Spencer–Szemerédi–Trotter improve upper to O(n^4/3) via incidence bound

1997

同上≤ ν ≤O(n^4/3)Sze

Székely's 1-page crossing-number proof of O(n^4/3)

2011

同上≤ ν ≤O(n log n)Mat

Matoušek: for most norms ν is near-linear — suggesting Euclidean "should" be too

2022

同上≤ ν ≤O(n^4/3) (constant ↓)ÁP

Ágoston & Pálvölgyi sharpen the constant in n^4/3 (only progress in 40 yrs)

2025

同上≤ ν ≤O(n log² n) genericABS / GST

Alon–Bucić–Sauermann + Greilhuber–Schildkraut–Tidor: ν ≤ (d/2 ± o(1)) n log² n for nearly all d-dim norms

2026-05

n^(1+δ), δ>0≤ ν ≤O(n^4/3)OpenAI

OpenAI (Chen + internal model, Sellke/Sawhney verifying): counterexample! ν(n) ≥ n^(1+δ) for infinitely many n — Erdős's 80-yr conjecture falls (δ implicit)

2026-05

n^(1+6·10⁻³⁸)≤ ν ≤O(n^4/3)Remarks

Remarks (9-author note, arXiv:2605.20695): human-digested, δ ≈ 6 × 10⁻³⁸ explicit; simplified to pro-2 tower + single split prime

2026-05

n^1.014114≤ ν ≤O(n^4/3)Sawin

Sawin solo (arXiv:2605.20579): drastic improvement, δ ≥ 0.014, within factor 24 of SST upper

The chart below plots all bounds on the exponent α = log ν / log n. This is the cleanest way to see "conjecture refuted": Erdős's conjectured upper is a curve decaying to 1 (1 + C/log log n); the new lower is a flat line (1 + δ); the flat line must eventually cross the decaying curve.

5.5 The other problem in Erdős 1946: distinct distances

Erdős's 1946 paper actually posed two dual discrete-geometry problems. One is unit distances ν(n) (this page's topic); the other is distinct distances D(n): the minimum number of distinct pairwise distances among n planar points.

Unit distances ν(n)

how often can ONE distance repeat?

Erdős conjecture

\leq n^{1 + o (1)}

best upper

O (n^{4/3})

(SST 1984)

best lower

n^{1.014}

(Sawin 2026)

statusconjecture false, big gap

Distinct distances D(n)

how few DISTINCT distances can occur?

Erdős conjecture

\geq c n / lo g n

upper (example)

O (n / lo g n)

(网格)

Guth–Katz 2015

\geq c \cdot n / lo g n

statusessentially solved (√log n gap)

The two problems are dual: ν asks "max repetition", D asks "min variety". Intuitively ν · D ≥ n²/2 (pigeonhole: n² pairs across D distances, the most common one occurs ≥ n²/(2D) times).

Historically D was the easier one: Guth–Katz 2015 used the polynomial method (an upgrade of Dvir's finite-field Kakeya argument) to prove $D (n) \geq c \cdot n / lo g n$ — only a $lo g n$ factor from Erdős's conjecture $n / lo g n$ .ν went the other way: now the lower bound is improved to n^1.014, but Erdős's conjectured direction (the upper bound side) was wrong. Litt in Remarks §6: "Solving D required new tools (polynomial method); refuting ν did not introduce new geometric tools, it just pushed number theory deeper — the difficulty profile is completely different."

6. Main theorem

Theorem 1.1 (OpenAI 2026)

There exists an absolute constant $δ > 0$ and infinitely many positive integers n such that $ν (n) \geq n^{1 + δ}$ .

This refutes Erdős's 1946 conjecture $ν (n) \leq n^{1 + C / l o g l o g n}$ : any constant-δ lower bound of the form $n^{1 + δ}$ eventually (for large enough n) exceeds $n^{1 + C / l o g l o g n}$ , because the latter's exponent 1 + C/log log n converges to 1.

It also refutes the stronger Erdős–Fishburn 1997 conjecture (every point has ≥ k equidistant neighbours ⟹ k ≤ n^o(1)): the construction's unit-distance graph has average degree n^Ω(1), and average-degree ≥ 2k always contains a subgraph of minimum degree ≥ k.

7. Classical ℤ[i] vs new L(i)

The r₂(k) trick in §4 is, geometrically, a "norm = k" factorisation in the Gaussian integers ℤ[i]. In ℤ[i], if k = q is a prime ≡ 1 mod 4, then q = π · π̄ for a Gaussian prime π. Multiplying many such q's gives many z ∈ ℤ[i] with z·z̄ = k — each z is a lattice point at distance √k from origin.

The new construction: replace ℚ by a larger totally real field L, and ℤ[i] by $O_{K}$ (K = L(i)). The non-trivial automorphism c (becoming complex conjugation under any embedding of K) translates "complex modulus 1" into "|σ(u)| = 1 for every complex embedding σ". Since [L:ℚ] → ∞, ℂ^f is genuinely high-dimensional and fits exponentially many norm-1 elements.

Classical 1946

Z [i] \subset C = R^{2}

dim2 (固定)

conjugationc(a + bi) = a − bi

norm-1 count

n^{c / l o g l o g n}

gives

ν (n) \geq n^{1 + c / l o g l o g n}

⇒

New 2026

O_{K} ↪ C^{f}, K = L (i), [L : Q] = f \to \infty

dim2f → ∞

conjugationc: K → K, σ(c·) = σ(·)̄

norm-1 count

\geq e^{γ f}

gives

ν (n) \geq n^{1 + δ}

7.5 Number-field glossary (8 terms to know before §8)

The 5-stage construction below uses these terms repeatedly. One sentence of intuition + one formal line each.

Totally real field

L ↪ R^{[L : Q]}

All conjugate embeddings of L land in ℝ. E.g.

Q (2)

Q (ζ_{7} + ζ_{7}^{- 1})

CM field

K = L (i), L totally real

Totally imaginary quadratic extension of a totally real field. The non-trivial automorphism c becomes ordinary complex conjugation under every complex embedding. This is what makes |σ(u)|=1 ⟺ u·c(u)=1.

Class number h(K)

h (K) = # Cl (K)

Size of fractional-ideals / principal-ideals. If 1, OK is a PID. Used as a pigeonhole denominator: 2^m ideals into h(K) classes ⇒ some class has ≥ 2^m / h(K) — this is where norm-1 elements come from.

Root discriminant rd(K)

rd (K) = ∣ D_{K} ∣^{1/ [K : Q]}

The n-th root of the discriminant — a "degree-independent" complexity. Unramified extensions preserve rd, the entire reason the Golod–Shafarevich tower is useful: degree → ∞ but rd stays bounded, so Minkowski's h(K) ≤ rd(K)^[K:ℚ] remains usable.

Frobenius element

Frob_{p} \in Gal (K / F)

The "lift" automorphism above an unramified prime p, acting as

x \mapsto x^{N p}

mod p. Frob class trivial ⟺ p splits completely. Chebotarev lets us prescribe these.

Frattini subgroup Φ(G)

Φ (G) = ⋂_{M max} M

Intersection of all maximal open subgroups. An element is in Φ(G) iff "removing it as a generator doesn't matter". Key: imposing Φ(G)-elements as relations doesn't drop d(G). That's why we can kill chosen Frobenius elements without breaking the Golod–Shafarevich count.

pro-p group

G = lim G_{n}, ∣ G_{n} ∣ = p^{k_{n}}

Inverse limit of finite p-groups — a topological group with all finite quotients p-groups. The Galois group of the maximal unramified pro-p extension F^{ur,p} is one; Golod–Shafarevich works in this category (r ≤ d²/4 ⇒ infinite).

Unramified extension

p O_{K} = P_{1} \dots P_{g}

Every prime p factors in K with all ramification indices = 1 (no repeated factors). Consequence: rd(K) stays put. Every layer of the OpenAI/Sawin tower is unramified, the reason rd doesn't blow up.

8. The construction in 5 stages

Click any stage to expand. The pipeline: base F → pro-3 tower → CM extension K = F(i) → Minkowski lattice → cut + project. The first three are the arithmetic part (build an algebraic object with high dim + many norm-1 elements); the last two are the geometric part (translate it into a planar point set).

9. Geometric part: lattice → polydisc → projection

This visualises §2 of the paper. The real proof has f → ∞; we drop to f = 2 (Gaussian integers ℤ[i] in ℂ) for intuition. Slide R to see |X| and unit-pair counts both grow exponentially — the gap between their exponents is what 1 + δ is.

The paper's key Lemma 2.4: some coset a + Λⱼ in W has ≥ $e^{γ f_{j} /2} ∣ X ∣$ ordered pairs whose difference lies in Uⱼ. That's why the projected planar set P has $ν (P) \geq \frac{1}{2} e^{γ f /2} ∣ P ∣$ unit pairs — far exceeding the trivial linear count.

10. The AI angle + references

From the paper — "Statement on AI Use"

"This problem was solved in a completely automated fashion. Our internal model was given an AI-written statement of the problem, and its output was sent to an AI grading pipeline, which indicated high confidence that the solution was correct. It was only after this point that internal human researchers and mathematicians began to examine the solution carefully. After preliminary AI-assisted verification and rewriting, a draft was sent to external mathematicians, including several number theory experts, who confirmed the proof's correctness (and have already simplified and strengthened the argument). The present manuscript is a human-edited exposition of the autonomously produced solution, with references, reorganized proofs, and additional explanatory material added afterward."

This is a milestone in AI for math: not mathematicians using AI as an assistant, but AI making the discovery end-to-end — refuting an 80-year-old conjecture — before any human got involved. The model's raw output is published alongside the paper. The OpenAI team: Lijie Chen used the internal model to generate the proof, while Mark Sellke and Mehtaab Sawhney verified correctness.

In May 2026, two groups of mathematicians rapidly followed up:

Remarks paper (Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, Wood; arXiv:2605.20695): human-digested version that computes the implicit δ > 0 explicitly to $δ \approx 6 \times 1 0^{- 38}$ , and simplifies the original pro-3 tower argument to pro-2 + single split prime (the simplification was suggested by Victor Wang).
Sawin solo (arXiv:2605.20579, 2026-05-20): drastic improvement to explicit $ν (n) \geq n^{1.014114} / C$ , i.e. $δ \geq 0.014$ — within a factor of about 24 of the SST upper bound $n^{4/3}$ (since $(4/3 - 1) /0.014 \approx 24$ ). Three improvements: work with any ideal (not just $O_{K}$ ), use the relative class number (not the full class number), quotient by a fixed power of Frobenius (not 1).

On whether an exact picture of the new construction exists: all three papers are non-constructive. The one explicit parameter choice (Remarks §2) — $T = {3, 5, 7, 11, 13, 17}$ , split prime 101, base field $L_{T} = Q (5, 13, 17, 21, 33)$ (degree 32 multi-quadratic) — only reaches the Golod–Shafarevich step; the individual layers $L_{j}$ are given by an existence theorem with no algorithmic description.

11. What the mathematicians said — quotes from Remarks §3-9

Noga Alon Princeton

"The solution of the problem by the internal model of OpenAI is, in my opinion, an outstanding achievement, settling a long-standing open problem. The fact that the correct answer is not n^(1+o(1)) is surprising, and the construction and analysis apply fairly sophisticated tools from algebraic number theory in an elegant and clever way."

Thomas Bloom Royal Society URF

"On April 16, I included this problem in a blog post on erdosproblems.com titled (tongue in cheek) 'Top 10 Erdős Problems'. The unit distance problem was the only one I included from discrete geometry. While I believed AI would make progress on at least a couple eventually, I did not expect this to happen just one month later!"

W. T. Gowers Cambridge / Collège de France, Fields

"If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that.… Even if it turns out that AI cannot yet find proofs that need long hint sequences, such proofs are very difficult to find for humans as well, so in the unlikely event that AI math progress stalls, we have probably entered an era where it will become difficult for humans to compete with AI at solving mathematical problems."

Daniel Litt Toronto / Sloan

"This is the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator. Analogues: Dvir's finite-field Kakeya proof; Huang's sensitivity conjecture proof. This unit-distance solution feels like the same flavor."

12. Open problems

The bound gap

Current: $n^{1.014} \leq ν (n) \leq O (n^{4/3})$ . Factor-24 gap in exponent. True value unknown. Could be 4/3, could be in between, could be smaller.

SST upper bound stuck

SST 1984 hasn't budged beyond constant tweaks. Can the upper be sharpened to $O (n^{4/3 - ε})$ ? Would be the biggest advance in 40 years.

Explicit construction?

All proofs rely on Golod–Shafarevich existence. Can we exhibit explicit P_j coordinates? Even one layer would be a milestone.

Erdős–Fishburn

Stronger variant: every point with ≥ k equidistant neighbours ⟹ k ≤ n^o(1). OpenAI also incidentally refuted this — the new construction has avg degree n^Ω(1).

Bloom's "Top 10" — which next?

2026-04 Bloom listed his 10 most prominent Erdős problems. Unit distance fell in a month. Which next? What distinguishes them? Does the "non-constructive AI counterexample" pattern generalise?

Higher dimensions

How about ν in ℝ^d, d ≥ 3? Sawin's method should generalise, but upper bounds (Erdős–Hickerson) are already Ω(n^{4/3}). Is the n^(1+δ) lower bound now reachable for d ≥ 3?

References

OpenAI 2026-05-20. Planar Point Sets with Many Unit Distances. Blog · PDF
Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, Wood 2026-05. Remarks on the Disproof of the Unit Distance Conjecture (human-digested, explicit δ ≈ 6·10⁻³⁸ + pro-2 simplification). arXiv:2605.20695
Sawin, W. 2026-05-20. An Explicit Lower Bound for the Unit Distance Problem (drastic improvement δ ≥ 0.014, within factor 24 of SST). arXiv:2605.20579
OpenAI 2026. Rewritten Chain of Thought for the Unit Distance Problem (raw model reasoning, published). PDF
Erdős, P. 1946. On sets of distances of n points. Amer. Math. Monthly 53(5):248–250.
Spencer, Szemerédi, Trotter 1984. Unit distances in the Euclidean plane.
Székely, L. 1997. Crossing numbers and hard Erdős problems in discrete geometry.
Guth, Katz 2015. On the Erdős distinct distances problem in the plane. Ann. of Math. 181(1):155–190.
Hajir, Maire 2001. Asymptotically good towers of global fields.
Golod, Shafarevich 1964. On the class field tower.
Alon, Bucić, Sauermann 2025. Unit and distinct distances in typical norms.
Greilhuber, Schildkraut, Tidor 2025. More unit distances in arbitrary norms.