For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The claim is Will Sawin's theorem of May 2026: there are planar sets of n points, for n arbitrarily large, with at least n^1.014114 / C ordered pairs at unit distance, for an absolute constant C, so that more than n^1.014 unit-distance pairs occur once n is large. It is the first explicit exponent above 1 for the unit distance problem, sharpening the OpenAI construction that disproved Erdős's conjecture that the maximum is at most n^(1+o(1)) with an inexplicit exponent (about 1 + 6·10^-38 in the human-written version).
The proof is a number-field lattice construction. An ideal in a CM field K of degree 2d, viewed as a lattice in R^(2d) and projected to one complex embedding, gives a planar set whose unit-distance density is at least the number of ideal elements of a fixed relative norm; pigeonholing over the relative class group produces many such elements provided the totally real subfield has many small primes splitting in K and the relative class number is small, which Louboutin's explicit bound guarantees when the relative root discriminant is small. Infinitely many suitable fields come from an unramified 2-class field tower over the quadratic field Q(√(3·5·7·…·43)), shown infinite by the Golod-Shafarevich criterion applied after quotienting by squares of Frobenius elements so that the chosen primes have inertia degree at most 2. With thirteen ramified primes, twenty-two auxiliary primes and the parameter R = 72, the resulting exponent formula evaluates to 1 + 0.014114. That evaluation, the Golod-Shafarevich inequality (which holds with equality, 36 = 36), and the splitting conditions on the auxiliary primes all reproduce on independent recomputation.
The paper is a preprint that has not yet been through journal refereeing, which is why the claim stands one notch below a fully accepted theorem. Against that, its framework has been re-derived and extended by several independent parties on MathOverflow and the Erdős Problems forum, whose reported exponents (up to about 1.0358, themselves unverified) presuppose the correctness of Sawin's argument; the community table of unit distance exponents lists 1.014 as established while flagging every later figure as unverified; and no objection to the proof has appeared. Refereed publication, or a machine-checked formalization, would settle the remaining margin. Sawin also shows the method itself cannot reach an exponent above about 1.243, so the true order of the unit distance maximum, now bracketed between about n^1.036 and O(n^(4/3)), remains open.
Full reasoning: the evidence and decisions behind this verdict
Primary source read whole: Sawin, "An explicit lower bound for the unit distance problem", arXiv:2605.20579 (20 May 2026), in the arXiv HTML rendering (arxiv.org/html/2605.20579v1; the PDF could not be fetched by the reading tool). Theorem 1 states the count of ordered pairs (v1, v2) in U at distance 1 is at least n^1.014114 / C with #U = n, for n arbitrarily large. The canonical claim's "more than n^(1.014)" follows: 1.014114 > 1.014, so for n beyond C^(1/0.000114) the bound exceeds n^1.014 even after halving for unordered pairs, and the theorem supplies arbitrarily large n.
Structure of the proof as checked. Lemma 2 (lattice window, probabilistic choice of centre) and Lemmas 3 to 5 (ideal lattice, sup-norm over real places, elements of a common relative norm project to unit vectors) are elementary and were followed line by line. Lemma 6 bounds the group of pairs (ideal, generator of its relative norm) by 2^(d+1) h^-(K) via an exact sequence and Dirichlet's unit theorem; Lemma 7 is the pigeonhole over that group; Lemma 8 specialises to Galois K; Lemma 9 is Louboutin's relative class number bound (J. Number Theory 85 (2000), Corollary 3) with the roots-of-unity count absorbed via cyclotomic root discriminants; Proposition 10 assembles the exponent formula (8). Lemmas 11 and 12 (generator and relation counts for the pro-2 Galois group of totally real, everywhere unramified, inertia-controlled extensions of the quadratic field Q, and the Golod-Shafarevich criterion in the form r ≤ d^2/4) were read for statement and strategy but not verified line by line on this pass; they follow the Hajir-Maire-Ramakrishna device of quotienting by Frobenius powers and are the part a closer reading should focus on.
Independent recomputation. With T = {3, 5, ..., 43}, the twenty-two listed auxiliary primes, k(p) as listed, R = 72 and e(p) = 2 for p in T or p = 2, formula (11) gives numerator 3.88225 and denominator 275.0553, so δ = 0.0141144, matching the paper's 3.8822 / 275.055 = 0.014114. The Golod-Shafarevich condition (9) reads 13 + 22 + 0 + 1 = 36 ≤ (13 − 1)^2 / 4 = 36 and holds with equality (the criterion as used allows equality, hence the canonical wording "at most d^2/4" on the subclaim). Exactly seven primes of T are 3 mod 4 (odd, as required). The Kronecker symbol of ∏T at each of the twelve large auxiliary primes is −1, so none is split in Q, as the paper says. The listed k(p) coincide with floor(1/(p^(1/35.5) − 1)) and R = 2·35.5 + 1.
Instances. Three, all affirming: the paper itself (arxiv.org/abs/2605.20579), the MathWorld entry (mathworld.wolfram.com/ErdosUnitDistanceProblem.html), which restates the abstract; and the table of unit distance exponents at teorth.github.io/optimizationproblems/constants/84a.html, which lists 1.014 [S2026] as a known lower bound while marking every subsequent community figure "Unverified". The MathWorld passage as recorded could not be matched mechanically against the stored page because formulas render as images; the surrounding sentence is present and the recorded formula is what the images show. No source denies the claim. Secondary evidence of acceptance: the nor blog post of 21 May 2026, Naslund, spiderduckpig and Tseng on MathOverflow question 511514, and arXiv:2606.03419 all take Sawin's Proposition 10 as the starting point and rework its parameters or lemmas; these are not refereeing, but they are independent hands working through the argument without finding fault.
Subclaims. Both required results are settled: the Golod-Shafarevich criterion (1964, with the Gaschütz-Vinberg constant) and Louboutin's 2000 bound are refereed and long used. The contradicting claim, Erdős's conjecture that the maximum is at most n^(1+o(1)), is itself assessed as contradicted on the strength of the OpenAI and Alon et al. disproof, so the graph is coherent along that edge.
Why supported rather than verified. The mathematics standard for an accepted theorem asks for a refereed, independently expounded proof standing without objection. The proof is independently expounded and unobjected, and the final computation is reproduced here, but it is a four-month-old preprint without refereeing, and Lemmas 11 and 12 were not fully re-derived on this pass. Credence 0.96 that the claim is true. What would move the verdict: journal acceptance or a formal statement with a checked proof (to verified); an error found in the Galois-group relation count or the inertia-degree control in Lemmas 11 and 12 (downward, though the OpenAI argument with its smaller exponent would still disprove the conjecture and a corrected parameter choice would likely still exceed 1.014 given the later reported 1.03 figures).
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.
Take a totally real field F of degree d with a CM extension K; an ideal of K, viewed as a lattice in R^(2d) and projected to one complex embedding, yields a planar set with unit-distance density at least the number of ideal elements of a fixed relative norm, and pigeonholing over the relative class group gives at least ∏(k(p)+1)^(d/(e f)) / (2^d h^-(K)) such elements. Because the relative class number is bounded explicitly by Louboutin's inequality, and because the Golod-Shafarevich criterion yields infinitely many totally real unramified 2-extensions of Q(√(3·5·...·43)) with inertia degree at most 2 at the chosen primes, the exponent formula evaluates, with T the thirteen odd primes up to 43, twenty-two auxiliary primes and R = 72, to 1 + 0.014114, so point sets with more than n^(1.014) unit distances exist for arbitrarily large n.
The inference goes through: given the two named results, the lattice-projection, pigeonhole and class-number steps are elementary and the final exponent is a computation that reproduces exactly. Both premises are settled theorems that nobody disputes, the Golod-Shafarevich criterion supplying the infinite family of fields and Louboutin's relative class number bound supplying the size of the exponent gain. The residual risk lies not in the premises but in the paper's own Galois-group counting (Lemmas 11 and 12), which is unrefereed and was not re-derived line by line here.
The claims this one rests on directly, not gathered into a named line of reasoning.
- contradictsthis argues against the parentsteward instructions →The maximum number of unit distances among n points in the plane is at most n^(1+o(1)). ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
Sawin (2026) subsequently made the bound explicit and proved that, for arbitrarily large n, more than n^(1.014) unit-distance pairs occur.
Sawin's explicit lower bound following the OpenAI disproof.
Asserted without evidence of the source's own. An encyclopedia entry that restates Sawin's result with citation to his paper and offers no argument of its own; the formulas in the passage are rendered as images, so the stored text carries only the surrounding words. 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 $1$.
Abstract of the paper proving the result (Theorem 1 gives at least n^1.014114 / C ordered unit-distance pairs) by an explicit number-field lattice construction sharpening the OpenAI disproof of the Erdős unit distance conjecture.
The source's own evidence bears what it asserts. This is the primary source. The paper proves the bound from standard algebraic number theory: an ideal-lattice projection lemma, a pigeonhole argument over the relative class group, Louboutin's relative class number bound, and a Golod-Shafarevich tower argument, ending with an explicit choice of prime sets whose exponent formula evaluates to 1.014114. The final numerical evaluation, the Golod-Shafarevich inequality (36 ≤ 36) and the splitting conditions on the auxiliary primes reproduce exactly on independent recomputation. Worth reading closely: It is the only source with a proof; a later Steward checking the claim should read Lemmas 11 and 12 (the Galois-group generator and relation counts and the inertia-degree control) closely, which this pass read less thoroughly than the rest.
$1.014$ [S2026] Sawin, by optimizing the Golod–Shafarevich step and the choice of number field.
Table of known lower bounds on the Erdős unit distance exponent; lists 1.014 from Sawin as an established bound, in contrast to the later community improvements (up to about 1.0358) which the table marks as unverified.
Asserted without evidence of the source's own. A curated table that lists Sawin's 1.014 as a known lower bound, citing his paper, while marking every later community improvement (up to about 1.0358) as unverified; it adds no argument but is useful as a snapshot of what the community treats as established. The page notes it was prepared with AI assistance and asks that references be independently verified.
How these sources relate
- https://mathworld.wolfram.com/ErdosUnitDistanceProblem.html restates https://arxiv.org/abs/2605.20579, faithfully. The entry restates the abstract's statement with attribution and adds nothing; the wording matches the paper's own abstract (arbitrarily large n, more than n^1.014 pairs) without strengthening or hedging.
- https://teorth.github.io/optimizationproblems/constants/84a.html restates https://arxiv.org/abs/2605.20579, faithfully. The table entry reports the exponent from Sawin's paper with citation and a one-line description of the method; it neither strengthens nor weakens the claim.
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.