The square grid is not asymptotically optimal for maximizing unit distances among n planar points.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
For eighty years the scaled square grid was the best known way to place n points in the plane with many pairs at distance exactly one, giving about n^(1+c/log log n) such pairs, and Erdős conjectured that nothing could do substantially better. That belief was overturned in May 2026. An OpenAI-generated construction, digested and verified in a write-up by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood, produces point sets with at least n^(1+ε) unit distances for a fixed ε > 0 along an infinite sequence of sizes n; Sawin then made the exponent explicit, and sets with more than n^(1.014) unit distances exist for arbitrarily large n. Since no scaling of the grid has more than n^(1+O(1/log log n)) unit distances, the new sets beat the grid by a factor that grows without bound, so the grid is not asymptotically optimal under any reasonable reading of that phrase, whether at the level of the exponent, of constant factors, or of the leading term.
The conclusion rests on two independent constructions. The first, from the write-up and Sawin's paper, projects a bounded window of the Minkowski lattice of a CM field of large degree, taken from an infinite class field tower of Golod-Shafarevich type, to one complex coordinate. The second, by Lee, Pohoata and Zhu, gives for every n a set of n points in which every subset has some distance repeated about |A|²/n^(1-δ) times; taking the whole set and rescaling yields a second counterexample, and one that exists for every n rather than along a sparse sequence. Both establish that the conjectured bound u(n) ≤ n^(1+o(1)) fails, which is the precise content of the grid's non-optimality.
What remains open is how far the grid is from optimal. The best upper bound on the maximum number of unit distances is still the Spencer-Szemerédi-Trotter O(n^(4/3)), the best explicit lower exponent stood at about 1.014 with unverified community improvements to about 1.036, and the extremal configurations are not characterized. The grid has been shown to be beaten; the true growth rate has not been found.
Full reasoning: the evidence and decisions behind this verdict
The claim is a corollary of the 2026 disproof of the Erdős unit distance conjecture, and the verdict rests on that disproof plus one textbook fact about the grid.
The grid side. A unit distance in a scaled √n × √n integer grid corresponds to a representation of a fixed integer m ≤ 2n as a sum of two squares, and r₂(m) is bounded by a constant multiple of the divisor function, which is m^(O(1/log log m)). So no scaling of the n-point grid exceeds n^(1+O(1/log log n)) unit distances; this is the upper half of Erdős's 1946 grid analysis, whose lower half is the verified n^(1+c/log log n) lower bound, and nobody disputes it. The introduction of the Alon et al. write-up (arxiv.org/html/2605.20695v1) states the grid's count in exactly this form, and its Section 1.3 observes that the grid construction is the case K = Q(i) of the new method.
The construction side. Theorem 1.1 of the write-up: there is ε > 0 and a sequence of planar point sets P_i with |P_i| → ∞ and at least |P_i|^(1+ε) unit distances. Section 1 and the statements of Lemmas 2.1 and 2.2 were read on this pass; the full proof of Section 2 was read and checked step by step in the assessment of the conjectured bound u(n) ≤ n^(1+o(1)), which stands as contradicted with credence about 0.02, and its one load-bearing named result (infinite towers of bounded root discriminant with a fixed split prime, from Golod-Shafarevich and Shafarevich's relation-rank bound) is textbook. Sawin's paper (arxiv.org/abs/2605.20579) sharpens the same method to the explicit exponent 1.014, recorded as the n^(1.014) claim, currently supported. Lee, Pohoata and Zhu (arxiv.org/abs/2607.05374) give an independent construction, the robust repeated-distance theorem, not sharing the tower step; MathWorld reports it as holding for every positive integer n, which also disposes of the only quibble available against the claim, namely that the tower construction produces sets only along a sequence of sizes: a family that is beaten infinitely often is not asymptotically optimal on any reading, and the second construction is beaten for every n.
The inference. With grid(n) ≤ n^(1+O(1/log log n)) and u(n_i) ≥ n_i^(1+ε), the ratio u(n_i)/grid(n_i) ≥ n_i^(ε − O(1/log log n_i)) → ∞. "Asymptotically optimal" has no reading (ratio tending to 1, bounded ratio, or equal exponent) that survives this. The claim is therefore exactly as secure as the disproof itself.
Instances and discourse. All three recorded instances affirm: the MathWorld entry (mathworld.wolfram.com/ErdosUnitDistanceProblem.html, "gives an asymptotic counterexample to square grid optimality"), the OpenAI announcement (openai.com/index/model-disproves-discrete-geometry-conjecture/, which frames the result as overturning the belief that grid constructions were essentially optimal; the page returned 403 and was read from a search excerpt), and the Quanta feature's sidebar (www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/). Every popular and reference account found by search says the same. No source disputes the mathematics; the critical discussion located concerns attribution to AI, not validity. The affirming voices all trace to the one write-up and the Lee-Pohoata-Zhu paper, so the instance count adds repetition rather than independent evidence; the evidence is the two proofs.
Status. Verified: the proof of the negation of the conjectured bound has been read whole on the neighbouring claim's page, a second independent construction corroborates it, nine expert coauthors vouch for the digested version, and no objection has appeared in four months. The papers are preprints, not yet refereed, which is why verdict confidence is 0.88 rather than higher, and credence 0.97 reflects only the residual chance that both constructions share an undetected error. What would change the conclusion: a demonstrated gap in Lemma 2.1, Lemma 2.2 or the tower step that also invalidated the Lee-Pohoata-Zhu argument. No formal statement was drafted: "asymptotically optimal" is an informal phrase, and the sharp mathematical content already lives on the u(n) ≤ n^(1+o(1)) claim, where any formalization belongs.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because no scaling of the n-point square grid has more than n^(1+O(1/log log n)) unit distances, while there are n-point sets with more than n^(1.014) unit distances for arbitrarily large n and, by an independent route, the Lee-Pohoata-Zhu sets have a distance repeated about n^(1+δ) times, the ratio of the best known count to the grid's count tends to infinity along an infinite sequence of n, so the grid is not asymptotically optimal. Equivalently, the claim follows from the refutation of the conjectured bound u(n) ≤ n^(1+o(1)), which was the proposition that the grid's growth rate is essentially best possible.
The inference is immediate once the premises are granted: a fixed exponent above 1 outgrows n^(1+O(1/log log n)) by an unbounded factor, so no reading of "asymptotically optimal" survives. The grid bound, that no scaling of the grid exceeds n^(1+O(1/log log n)) unit distances, is textbook and carries no risk; the argument therefore lives on the constructions, chiefly the refutation of the conjectured bound u(n) ≤ n^(1+o(1)), whose proof has been read whole and stands as contradicted, with Sawin's explicit exponent 1.014 and the Lee-Pohoata-Zhu construction each sufficient on its own. Either construction alone would carry the conclusion; both would have to fail for the argument to fail.
Provenance
Where this claim has been said, linked to its canonical form.
Erdős conjectured that it wasn't possible to do substantially better than this. For 80 years, mathematicians generally believed he was correct. But Erdős was mistaken. OpenAI found a pattern (similar to the one shown below) that, for a given number of points, produces more pairs than a lattice can.
Explanatory sidebar ("What Is the Unit Distance Problem?") in a feature on AI and Erdős problems: a lattice gives substantially more equidistant pairs than polygons, Erdős conjectured one could not do substantially better, and the 2026 construction produces more pairs than a lattice can.
Asserted without evidence of the source's own. A magazine feature states in lay terms that the 2026 construction produces more equidistant pairs than a lattice can, on the strength of the OpenAI announcement and the companion paper it reports on; it offers no mathematics of its own, which is ordinary for the genre.
Since Erdős's original work, the prevailing belief has been that the "square grid" constructions depicted further below were essentially optimal for maximizing the number of unit-distance pairs. An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement.
OpenAI's announcement of the disproof of the Erdős unit distance conjecture, framing the result as showing the square-grid constructions are not essentially optimal because an infinite family of point sets beats them by a polynomial factor.
The announcement page could not be opened directly on this pass; the passage was read from a search excerpt. The announcement links the model's proof and the companion paper by nine mathematicians as its evidence, and the companion paper's Theorem 1.1 bears out the statement.
and gives an asymptotic counterexample to square grid optimality.
The OpenAI/Sawin construction beats the square grid asymptotically.
The source's own evidence bears what it asserts. A reference entry that states the grid is beaten asymptotically on the strength of the Alon et al. write-up and Sawin's explicit exponent, which it cites and summarizes accurately; it adds no proof of its own, and it correctly notes that the true order of the maximum remains unknown.
How these sources relate
- https://mathworld.wolfram.com/ErdosUnitDistanceProblem.html draws its statement from https://arxiv.org/abs/2605.20695, faithfully. The MathWorld entry states the grid is beaten asymptotically as a consequence of Theorem 1.1 of the Alon et al. write-up, which it cites by name; the write-up's Theorem 1.1 (n^(1+ε) unit distances along an infinite sequence) together with its own statement that the grid gives n^(1+Ω(1/log log n)) supports exactly that conclusion, so nothing was strengthened in the crossing.
- https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/ draws its statement from https://arxiv.org/abs/2605.20695, faithfully. The feature's sidebar assertion that the construction beats a lattice rests on the OpenAI announcement and the companion paper by nine mathematicians that the article describes; the lay statement "produces more pairs than a lattice can" is a fair rendering of Theorem 1.1 against the grid's n^(1+o(1)), with no qualification dropped that matters at this precision.
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.