OP-12099OpenCombinatorics

Erdős unit distance exponent ( $C_{84a}$ )

Erdős unit distance exponent: determine $C_{84a}$ or improve its bounds; the repository records 1.014 (1.03583*) (lower) and $\frac{4}{3}\approx 1.3333$ (upper).

MSC 52C10 · Erd?s problems and related topics of discrete geometry

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

For a finite set $P \subset \mathbb{R}^2$ of $n$ points, let

u(P) := \lvert \{ \{x,y\} \subset P : \lvert x-y \rvert = 1 \} \rvert

denote the number of unordered pairs of points in $P$ at unit distance, and let $u(n) := \max_{\lvert P\rvert = n} u(P)$ denote the maximum number of unit distances among any set of $n$ points in the plane. The Erdős unit distance exponent is

C_{84} := \limsup_{n \to \infty} \frac{\log u(n)}{\log n}.

Equivalently, $C_{84}$ is the infimum of all $\alpha$ for which $u(n) \ll_\alpha n^\alpha$ as $n \to \infty$ .

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].

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Background

The unit distance problem is one of the most well-known problems of Erdős in combinatorial geometry. The bound $u(n) \ll n^{4/3}$ of [SST1984] has not been improved in its exponent; subsequent work has only sharpened the implicit constant (see e.g. [AS2020]).
The OpenAI announcement [O2026] is the first widely publicized example of an autonomous AI system producing a one-shot disproof of a long-standing open conjecture in mathematics.
The same techniques apply to the related problem of constructing point sets with many distinct pairs at any fixed algebraic distance; see [ABGLSSTWW2026] for discussion.
A closely related quantity is the distinct distance exponent $\liminf \log d(n)/\log n$ , where $d(n)$ is the minimum number of distinct distances determined by $n$ points in the plane; this was shown to equal $1$ by Guth and Katz (2015).
The reference [S2026] also provides a "bound on the bound" that shows that the limit of the "lattice and pigeonhole in a number field method" is at most $1 + \frac{1}{4.116} \approx 1.243$ .

Related constructions can be used for the sum-product problem: see the page for $C_{84b}$ .

References (from the repository page)

[E1946] Erdős, Paul. On sets of distances of $n$ points. American Mathematical Monthly 53 (1946), 248–250. DOI: 10.2307/2305092.
[SST1984] Spencer, Joel; Szemerédi, Endre; Trotter, William T. Unit distances in the Euclidean plane. In: Graph Theory and Combinatorics (Cambridge, 1983), pp. 293–303, Academic Press, London, 1984.
[AS2020] Ágoston, Péter; Pálvölgyi, Dömötör. An improved constant factor for the unit distance problem. arXiv:2006.06285 (2020).
[O2026] OpenAI. An OpenAI model has disproved a central conjecture in discrete geometry. Blog post, 20 May 2026. openai.com/index/model-disproves-discrete-geometry-conjecture.
[ABGLSSTWW2026] Alon, Noga; Bloom, Thomas F.; Gowers, W. T.; Litt, Daniel; Sawin, Will; Shankar, Arul; Tsimerman, Jacob; Wang, Victor; Wood, Melanie Matchett. Remarks on the disproof of the unit distance conjecture. arXiv:2605.20695 (2026).
[S2026] Sawin, Will. An explicit lower bound for the unit distance problem. arXiv:2605.20579 (2026).
[EPF90] Bloom, Thomas F. (ed.). Erdős Problem #90: Discussion thread. erdosproblems.com/forum/thread/90 (accessed 22 May 2026).
[MO511514] What is the unit distance exponent? MathOverflow question 511514, mathoverflow.net/q/511514 (accessed 22 May 2026).

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.014 (1.03583*); best known upper bound $\frac{4}{3}\approx 1.3333$ . Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$3/2$	[E1946]	Erdős' original argument, based on the observation that the unit-distance graph contains no $K_{2,3}$ .
$4/3 = 1.333\dots$	[SST1984]	Spencer, Szemerédi and Trotter; via the Szemerédi–Trotter incidence theorem.

Known lower bounds

Bound	Reference	Comments
$1$	[E1946]	A $\sqrt{n} \times \sqrt{n}$ portion of the integer lattice gives $u(n) \gg n^{1 + c/\log\log n}$ , hence $\limsup \log u(n)/\log n \geq 1$ . Conjectured by Erdős to be sharp.
$> 1$ (inexplicit)	[O2026], [ABGLSSTWW2026]	An OpenAI internal reasoning model produced a (one-shot) construction giving $u(n) \gg n^{1+\varepsilon}$ for some $\varepsilon > 0$ and infinitely many $n$ , disproving Erdős' conjecture. The construction is number-theoretic, using algebraic number fields of large degree and small discriminant via a Golod–Shafarevich argument. A particular parameter choice in [ABGLSSTWW2026] yields the explicit value $\varepsilon \approx 6.24 \cdot 10^{-38}$ .
$1.014$	[S2026]	Sawin, by optimizing the Golod–Shafarevich step and the choice of number field.
$1.03184\dots$	[EPF90]	mlewko, posted on the Erdős Problems forum (21 May 2026); reuses Sawin's argument with optimized finite data ( $\lvert T\rvert = 74$ , $\lvert S\rvert = 1082$ ) produced by ChatGPT 5.5 Pro. Unverified.
$1.03188\dots$	[MO511514]	spiderduckpig, posted on MathOverflow (21 May 2026); replaces the primes $347,353$ in mlewko's $T$ by $503,601$ . Unverified.
$1.03190\dots$	[MO511514]	spiderduckpig, MathOverflow comment (22 May 2026); further replaces $337,349,373$ in $T$ by $389,443,587$ . Unverified.
$1.0333487\dots$	[MO511514]	spiderduckpig, MathOverflow comment (22 May 2026); claimed via a weighted-Zassenhaus-filtration modification of Sawin's Lemma 11, so requires a change to the proof structure and not just finite data. Unverified.
$1.03447\dots$	[MO511514]	Naslund, MathOverflow answer (22 May 2026); combines spiderduckpig's Golod–Shafarevich refinement with three further structural improvements to Sawin's argument — a Louboutin-style bound on $L(1,\chi)$ via $\zeta_{E_T}$ , a sphere-overlap refinement of Lemmas 4–5, and a narrow-class-group sharpening of Lemma 7 — then re-optimizes $T$ (the first $51$ odd primes minus $\{227,233\}$ ). Hybrid ChatGPT 5.5 Pro / human work. Unverified.
$1.0346749\dots$	[MO511514]	Naslund, revised (≈24 May 2026); corrects the optimization to fully include the narrow-class-group improvement, with optimizer $\lvert T\rvert = 48$ . (An intermediate $T$ -tweak had given $1.03448\dots$ .) Unverified.
$1.03158935$	[MO511514]	Tseng, MathOverflow answer (25 May 2026). Does not improve the record exponent, but supplies a reproducible pointwise certificate: $u(n) \geq \frac{1}{128 P_{89}} n^{1.03158935}$ for all sufficiently large $n$ , where $P_{89}$ is the product of the first $89$ odd primes. Uses Sawin's construction as a black box plus prime-ideal microselection, a fractional-knapsack model and a dyadic-circle covering certificate (repository). AI-assisted. Unverified.
$1.0346609571815997\dots$	[MO511514]	spiderduckpig, revised answer (25 May 2026); a reproducible finite-data/certificate improvement. Does not exceed Naslund's $1.0346749\dots$ . Unverified.
$1.03583\dots$	[MO511514]	Naslund, revised (≈25 May 2026); replaces the sphere-packing estimate by a covolume bound on the ideal lattice (asserted to be essentially optimal for this part of the argument), re-optimizing to $\lvert T\rvert = 39$ . Hybrid ChatGPT 5.5 Pro / human work. Current best. Unverified.

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Erdős unit distance exponent ($C_{84a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/84a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Record

Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Erdős unit distance exponent ($C_{84a}$)
Added: 5 Nov 2026
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Fields

Combinatorics