The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Assessment
Available evidence weighs against the claim.
This is the Erdős unit distance conjecture, posed in 1946: that n points in the plane determine at most n^(1+o(1)) pairs at unit distance, so that the rescaled square grid, which gives about n^(1+c/log log n), is essentially the best possible. Erdős offered $500 for a proof or disproof, and for eighty years the conjecture was widely believed; the upper bound stood at O(n^(4/3)) (Spencer, Szemerédi and Trotter, 1984) and no construction beat the grid.
The conjecture is false. In May 2026 an OpenAI reasoning model produced a construction, written up and verified by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood, giving an infinite family of planar point sets with at least n^(1+ε) unit distances for a fixed ε > 0. The construction takes CM number fields of growing degree from an infinite class field tower of bounded root discriminant in which a fixed prime splits completely (a consequence of the Golod-Shafarevich theorem, the existence of such towers), uses a pigeonhole argument in the class group to produce exponentially many elements of absolute value 1 in every embedding, and projects a bounded window of the ring of integers to the plane. The exponent from the simplest parameters is minuscule (about 1 + 6·10^(-38)), but Sawin immediately optimized the same method to show more than n^(1.014) unit distances for arbitrarily large n, and community refinements have pushed the exponent to about 1.036. Lee, Pohoata and Zhu then gave an independent robust construction that yields a second counterexample by a different route.
No credible objection to the disproof has appeared; the proof is short, rests on standard algebraic number theory, and has been checked by several of the leading mathematicians in the area. What remains open is the true order of growth: the maximum now lies somewhere between about n^(1.036) and O(n^(4/3)), and Sawin has shown the number-field method itself cannot exceed roughly n^(1.243).
Full reasoning: the evidence and decisions behind this verdict
The verdict rests on a direct reading of the proof of the negation, not on report alone. Section 2 of "Remarks on the disproof of the unit distance conjecture" (arxiv.org/html/2605.20695v1) was read whole. Lemma 2.1: for a δ-separated full-rank lattice Λ in C^f whose projection to one coordinate is injective, the points of a translate inside the polydisc of radius R, projected to that coordinate, give at least |U_Λ|·|(a+Λ) ∩ B_(R−1)|/2 unit-distance pairs (translation by any lattice vector with all coordinates of modulus 1 moves a point of B_(R−1) into B_R and gives a unit distance in the projection) among at most (9R²/δ²)^f points (a packing bound). Lemma 2.2: pigeonholing the ideals ∏ P_j^(a_j) P̄_j^(k_j − a_j) in the class group yields at least ∏(k_j+1)/h(K) principal ideals (α) with αᾱ a unit, and u = α/ᾱ then has modulus 1 in every embedding of a CM field and lies in Q^(−2) ⊆ D^(−1)O_K. Combining: with root discriminant bounded by r, the class number bound h_K ≤ |Disc K| ≤ r^(2f) gives u = (k+1)/r² elements per coordinate direction against skewness v = r/2, and choosing k+1 = ⌈18r³/π⌉ makes u > 36v/π, so log(2ν(P))/log|P| is bounded below by a constant greater than 1 as f → ∞. The tower step, that the maximal pro-2 extension of Q unramified outside {3,5,7,11,13,17} and completely split at 101 is infinite, follows from Golod-Shafarevich with Shafarevich's relation-rank bound (Koch, Theorems 11.5, 11.8); this is recorded as the argument's one load-bearing named result, the existence of infinite towers of bounded root discriminant with a fixed split prime, and is textbook. Each step checked; no gap found. The claim as stated says u(n) ≤ n^(1+ε) for every ε > 0 and all large n; an infinite family with u(n_i) ≥ n_i^(1+ε₀) for fixed ε₀ > 0 refutes it outright, so the theorem is precisely the negation of this claim.
Corroboration: Sawin's paper (arxiv.org/abs/2605.20579) states an explicit exponent 1.014 from the same method; this is a separate claim already in the graph, read here only at the abstract. Lee, Pohoata and Zhu (arxiv.org/abs/2607.05374) give a robust repeated-distance construction whose whole-set case is a second counterexample by a different method; only the abstract was read, so this line is corroborating rather than independently verified here. The reference page on the unit distance exponent (teorth.github.io/optimizationproblems/constants/84a.html) and the MathWorld entry both record the conjecture as disproved and add nothing beyond the papers. A search for objections to the validity of the disproof found none; the critical commentary located concerns the AI-attribution narrative, not the mathematics.
Instance set: every recorded source denies the claim (MathWorld, the Alon et al. write-up, Sawin, the exponent reference page, Lee-Pohoata-Zhu). The MathWorld instance had been recorded with the wrong stance and was corrected after reading the source; its recorded passage differs from the stored page text only because the page renders formulas as images, so the mechanical quote check fails while the prose matches. The Alon et al. instance's passage (Theorem 1.1) sits in the HTML full text rather than the abstract page recorded as its URL. The affirming voice in the discourse is historical: Erdős conjectured the bound and repeated it for decades, and expositions before May 2026 stated it as the expected truth; no post-disproof source affirms it. The erdosproblems.com page for Problem 90 could not be fetched (403).
Credence that the claim is true is about 0.02, reflecting only the residual chance that an error survived in a short proof checked by nine expert authors and reproduced by an independent construction. Verdict confidence 0.93. What would change the conclusion: a demonstrated gap in Lemma 2.1, Lemma 2.2, or the tower existence step that also invalidated Sawin's and Lee-Pohoata-Zhu's arguments. Another pass would add little; the remaining useful work is checking the Lee-Pohoata-Zhu proof directly or a formalization, and the open question of the true exponent belongs to other claims.
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 infinite towers of number fields with bounded root discriminant and a fixed completely split prime exist, one can take CM fields K of growing degree f from such a tower, use the many prime ideals above the split prime and a pigeonhole argument in the class group to produce exponentially many (in f) elements of K of absolute value 1 in every embedding with bounded denominator, and project a bounded window of the Minkowski lattice of K to one complex coordinate; since the number of unit-distance pairs grows like c^f with a base exceeding the base governing the number of points, the resulting planar sets have at least n^(1+ε) unit distances for infinitely many n, contradicting the claim. Optimizing the tower and the parameters makes this explicit: For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs..
The inference goes through: the two lemmas (geometry-of-numbers count and class-group pigeonhole) are elementary once the fields are supplied, and the exponent computation is explicit. The argument rests entirely on the existence of infinite towers of number fields with bounded root discriminant and a fixed completely split prime, a settled consequence of the Golod-Shafarevich theorem with Shafarevich's relation bounds. Sawin's explicit exponent is the same construction optimized and adds quantitative force without being needed for the refutation itself.
Because infinite towers of number fields with bounded root discriminant and a fixed completely split prime exist, one can take CM fields of growing degree from such a tower, use the many prime ideals above the split prime together with a pigeonhole argument in the class group to obtain exponentially many elements of absolute value 1 in every embedding with bounded denominator, and project a bounded window of the field's Minkowski lattice to one complex coordinate; the unit-distance pairs then grow with a larger exponential base than the number of points, so the resulting planar sets have at least n^(1+ε) unit distances for infinitely many n, which is incompatible with the conjectured n^(1+o(1)) bound. Optimizing the tower and parameters makes this explicit: For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs..
The inference goes through: once the fields are supplied, the geometry-of-numbers count and the class-group pigeonhole are elementary, and the exponent computation is explicit. The argument rests entirely on the existence of infinite towers of number fields with bounded root discriminant and a fixed completely split prime, a settled consequence of the Golod–Shafarevich theorem with Shafarevich's relation bounds. Sawin's explicit exponent is the same construction optimized and adds quantitative force without being needed for the refutation itself.
Given There exist n-point planar sets where every subset has a distance repeated at least c|A|^2/n^(1-delta) times., taking the subset to be the whole n-point set yields one distance occurring about n^(1+δ) times; rescaling that distance to 1 gives, for every n, an n-point planar set with about n^(1+δ) unit distances, which is incompatible with u(n) <= n^(1+o(1)).
Granting the premise, the inference is immediate: applying the subset bound to the whole set gives one distance repeated about n^(1+δ) times, and rescaling makes it the unit distance. The argument therefore stands or falls with the robust repeated-distance theorem of Lee, Pohoata and Zhu, whose proof has not yet been examined on this claim's page; it is currently corroboration of a refutation already established by the number-field construction rather than the primary ground for it.
Given There exist n-point planar sets where every subset has a distance repeated at least c|A|^2/n^(1-delta) times., taking the subset to be the whole n-point set yields one distance occurring about n^(1+δ) times; rescaling that distance to 1 gives, for every large n, an n-point planar set with about n^(1+δ) unit distances, which is incompatible with the conjectured n^(1+o(1)) bound.
Granting the premise, the inference is immediate: applying the subset bound to the whole set gives one distance repeated about n^(1+δ) times, and rescaling makes it the unit distance. The argument stands or falls with the robust repeated-distance theorem of Lee, Pohoata and Zhu, whose proof has not been examined here; it is corroboration of a refutation already established by the number-field construction rather than the primary ground for it.
Provenance
Where this claim has been said, linked to its canonical form.
Every source on record denies the conjectured bound, and all of them trace to one event: the May 2026 disproof. The primary evidence is the complete proof in the write-up by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood, which the MathWorld entry and the reference page on the unit distance exponent restate faithfully without adding evidence; Sawin's explicit-exponent paper shares an author and a method with the write-up, so those two are one voice sharpened rather than two. The one independent line is the Lee, Pohoata and Zhu construction, which reaches the same conclusion by a different route. A reader should open the write-up's Section 2 first, since the proof there is short enough to check directly.
Two months ago today (20th May 2026), ChatGPT disproved Erdős’ Unit Distance conjecture in discrete geometry.
Opening of the 'Unit distance' section describing an AI-generated disproof later formalized in Lean.
Asserted without evidence of the source's own. The post reports the disproof rather than proving it, relying on the announcement, on the testimony of mathematicians who checked the argument, and on a complete Lean formalization the author inspected in June 2026 and describes as proving the counterexample from the axioms of mathematics alone. Its distinctive contribution is the report of that formal check; for the mathematics itself a reader should open the human-verified write-up.
The conjectured matching bound, in the form u(n)<=n^(1+o(1)), where o(1) denotes a quantity tending to 0 in little-O notation, or more concretely u(n)<=n^(1+C/lnlnn) for some absolute constant C, was refuted by an OpenAI-generated proof (Alon et al. 2026, OpenAI 2026).
On the Erdős unit distance conjecture and its refutation.
The source's own evidence bears what it asserts. A reference entry that restates the 2026 disproof accurately, crediting the OpenAI-generated proof, the Alon et al. write-up, and Sawin's explicit exponent; it adds no evidence of its own beyond finite illustrative pieces of the construction, and correctly notes that the true order of the maximum remains unknown. The quoted passage was not found in the stored copy of this source.
There exists ε > 0 such that the following holds. There exists a sequence of point sets P_i in R^2 such that |P_i| → ∞ and the number of unit distances in P_i is at least |P_i|^{1+ε} for all i.
Theorem 1.1 of a human-verified, digested write-up of the OpenAI-generated counterexample; the theorem is the negation of the conjectured bound u(n) <= n^(1+o(1)), and the paper is titled as its disproof.
The source's own evidence bears what it asserts. The primary mathematical record of the disproof. Section 2 gives a full proof from two lemmas (a geometry-of-numbers count and a class-group pigeonhole) plus the Golod-Shafarevich existence of suitable towers, with an explicit if tiny exponent; the authors describe the original AI proof as valid and their version as simplified and generalized. The evidence is the proof itself, which is short enough to check. Worth reading closely: Section 2 is the complete proof; a Steward reassessing should verify Lemmas 2.1 and 2.2 and the tower step directly rather than rely on the authors' standing. The quoted passage was not found in the stored copy of this source.
We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the same result with an inexplicit exponent greater than $1$, drastically improving on the best previous lower bound and disproving a conjecture of Erdős.
Abstract of a paper making the disproof explicit with exponent 1.014.
The source's own evidence bears what it asserts. Sawin, a coauthor of the verified write-up, sharpens the same number-theoretic method to an explicit exponent; the abstract was read, and the result is independently restated by the reference page on the unit distance exponent, which also lists later community improvements to about 1.036 that remain unverified.
The problem was introduced by Erdős in 1946 [E1946], who conjectured that $u(n) = n^{1+o(1)}$, i.e. $C_{84} = 1$. This conjecture was disproved in May 2026 by an OpenAI internal model [O2026], with a human-verified writeup given in [ABGLSSTWW2026]; an explicit improved lower bound was obtained shortly afterwards by Sawin [S2026].
Reference page for the unit distance exponent, tabulating known upper and lower bounds; states the conjecture disproved and lists subsequent explicit exponents up to about 1.0358 (the later ones marked unverified).
The source's own evidence bears what it asserts. A curated reference page that restates the disproof on the basis of the Alon et al. write-up and Sawin's paper, and is careful to mark the later community exponents as unverified. It adds no proof of its own.
Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture.
Abstract of a paper giving a robust repeated-distances construction that yields a second counterexample to the conjectured bound.
The source's own evidence bears what it asserts. An independent construction by different authors, inspired by the number-field counterexample but proceeding through a robust Ramanujan-type estimate; the counterexample follows by taking the whole set. Only the abstract was read on this pass; the proof itself was not checked here. Worth reading closely: This is the only line of refutation not sharing the Golod-Shafarevich tower method; checking its proof would confirm the disproof rests on two independent constructions rather than one.
There exists ε > 0 such that the following holds. There exists a sequence of point sets P_i in R^2 such that |P_i| → ∞ and the number of unit distances in P_i is at least |P_i|^{1+ε} for all i.
Theorem 1.1 of the human-verified, digested write-up of the OpenAI-generated counterexample; the paper is titled as the disproof of the conjecture and gives a complete proof of the negation in Section 2.
The source's own evidence bears what it asserts. The primary mathematical record of the disproof. Section 2 proves the theorem from two lemmas, a geometry-of-numbers count and a class-group pigeonhole, plus the existence of Golod–Shafarevich towers with a split prime; the authors describe the original AI proof as valid and their version as simplified and generalized. The evidence is the proof itself, which is short enough to check. Worth reading closely: Section 2 is the complete proof; a Steward reassessing should verify Lemmas 2.1 and 2.2 and the tower step directly rather than rely on the authors' standing. The quoted passage was not found in the stored copy of this source.
We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the same result with an inexplicit exponent greater than $1$, drastically improving on the best previous lower bound and disproving a conjecture of Erdős.
Abstract of a paper making the disproof explicit with exponent 1.014 via an optimized Golod–Shafarevich construction.
The source's own evidence bears what it asserts. Sawin, a coauthor of the verified write-up, sharpens the same number-theoretic method to an explicit exponent; only the abstract was read on this pass. It corroborates the disproof but shares its method and an author with the write-up, so it is the same voice made explicit rather than an independent one.
An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement. The proof has been checked by a group of external mathematicians.
Announcement of the result; states the conjecture disproved by an infinite family of point sets with polynomially more unit distances than the grid, and that external mathematicians checked the proof.
Asserted without evidence of the source's own. The announcement states the result and that external mathematicians checked it; the proof itself lives in the nine-author write-up. Read here only in excerpt.
An internal AI model — one not available to the public — had come up with a counterexample to the “unit distance” problem, a conjecture made in 1946 by Paul Erdős
Sidebar: "Erdős conjectured that it wasn't possible to do substantially better than this... But Erdős was mistaken. OpenAI found a pattern... that, for a given number of points, produces more pairs than a lattice can."
How these sources relate
- https://mathworld.wolfram.com/ErdosUnitDistanceProblem.html draws its statement from https://arxiv.org/abs/2605.20695, faithfully. MathWorld states the refutation on the basis of the Alon et al. write-up (and the OpenAI proof file), summarizing Theorem 1.1 accurately: an infinite family with n^(1+ε) unit distances for fixed ε. Nothing is strengthened or dropped.
- https://teorth.github.io/optimizationproblems/constants/84a.html draws its statement from https://arxiv.org/abs/2605.20695, faithfully. The page states the disproof on the authority of the Alon et al. write-up and reproduces its explicit exponent exactly as given in equation (2.2) of that paper.
- https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/ restates https://arxiv.org/abs/2605.20695, faithfully. The post restates the disproof on the strength of the announcement and the checking mathematicians' write-up (the nine-author remarks), adding no mathematical evidence of its own beyond the later Lean formalization report. Its summary of the proof structure (Golod–Shafarevich towers yielding the counterexample) matches the write-up's Section 2.
- https://arxiv.org/abs/2605.20579 and https://arxiv.org/abs/2605.20695 share an author. Will Sawin is sole author of the explicit lower bound paper and one of the nine authors of the Remarks write-up; both bylines were read on arXiv.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.