Erdős unit distance exponent

Description of constant

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].

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.
$4/3 - \varepsilon$	[OAI2026]	OpenAI (23 September 2026). Proves $u(n) = O(n^{4/3-\varepsilon})$ for an absolute constant $\varepsilon > 0$, which is not made explicit.

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. Unverified.
$1.0427$*	[Nas2026], [NasCode2026], [Pal2026]	Naslund (October 2026). Gives $u(n) \geq n^{1.0427}$ for arbitrarily large $n$, using a tower over $\mathbb{Q}(\sqrt{241})$, quadratic extensions of mixed signature and averaging over norm-one units. The registered Lean 4 theorem proves the exponent $10427/10000$ conditional on one explicit zeta inequality for a field of degree $512$; a separate interval computation bounds its left side by $0.0848335193 < 0.0852$. See the verification details below.
$1.04317$*	[Nas2026b], [NasCode2026], [Pal2026b]	Naslund (October 2026). Gives $u(n) \geq n^{1.043171}$ for arbitrarily large $n$, refining the construction of the $1.0427$ entry: an infinite pro-$2$ tower over $\mathbb{Q}(\sqrt{241})$ ramified only above $2$, $3$ and $5$, with Frobenius conditions at the two primes above $29$ and at one prime above $41$. The relative zeta value is bounded through the $27$ dihedral quotients of order $8$ of the tower group, a census of split primes up to norm $4\cdot 10^{13}$, and the $576$ $L$-functions of degrees $4$ and $8$ of a field of degree $8192$, evaluated rigorously at $21$ real points near $1$ and combined through a kernel inequality with coefficients of both signs. Computer-assisted. Version 2 of the Palomar registration proves in Lean the slightly smaller exponent $20863/20000 = 1.04315$ conditional on one explicit zeta inequality $H_W$, and a separate interval computation bounds its left side by $0.0509171473 < 0.050969$. See the verification details below. Current best recorded bound.

Additional comments and links

Asterisks on the two October rows. The Palomar registrations cover less than the recorded values: version 2 proves $20863/20000 = 1.04315$ in Lean, and only conditionally on an explicit zeta inequality $H_W$ whose verification is a Python receipt outside Lean. The recorded $1.04317$ therefore rests partly on unformalized computation, so both rows carry the marker.
For the $1.04317$ bound, version 2 of the Palomar registration [Pal2026b] registers the implication from an explicit hypothesis $H_W$, an inequality for the Dedekind zeta function of a field $E_W$ of degree $8192$ generated by square roots of four explicit elements over the field of degree $512$, to finite planar sets $U_j$ with $\lvert U_j\rvert\to\infty$ and $u(U_j)/\lvert U_j\rvert^{1.04315}\to\infty$. Apart from $H_W$, the Lean proof uses only the standard axioms propext, Quot.sound and Classical.choice. $H_W$ is established outside Lean: papers/0.043171/certificates/dihedral/h_w_receipt.py in [NasCode2026] (at commit b43f0c5) bounds its left side by $0.0509171473$, below the threshold $0.050969$, from the certified $L$-values of [Nas2026b]. The exponent $1.043171$ itself rests on the computer-assisted certificate of [Nas2026b], which goes beyond the formalized statement. The manuscript and formalization disclose substantial AI use.
For the $1.0427$ bound, [Pal2026] registers the implication from the explicit hypothesis H241 to finite planar sets $U_j$ with $\lvert U_j\rvert\to\infty$ and $u(U_j)/\lvert U_j\rvert^{1.0427}\to\infty$. The Lean proof uses only the standard axioms propext, Quot.sound and Classical.choice. H241 itself is established outside Lean: papers/0.04273/certificates/h241_receipt.py in [NasCode2026] evaluates the zeta expression using the $255$ quadratic Hecke $L$-functions of the field of degree $512$ and an Euler product with an explicit tail. At commit 14289d9, its certified upper bound is $0.0848335192604133$, below $0.0852$. Thus the numerical certificate is a separate part of the proof, beyond the implication checked by Palomar. The manuscript and formalization disclose substantial AI use.
The unit distance problem is one of the most well-known problems of Erdős in combinatorial geometry. The exponent of the bound $u(n) \ll n^{4/3}$ of [SST1984] was first improved in [OAI2026]; earlier work 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

[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).
[OAI2026] OpenAI. A power saving for planar unit distances. OpenAI Math Release preprint, github.com/openai/math (23 September 2026).
[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).
[Nas2026] Naslund, Eric. An Exponent of 1.04273 for the Unit Distance Problem. Hexagon:2610.00016v1 (PDF), 2 October 2026; manuscript dated 30 September 2026. Theorem 1.1 and Remark 4.40.
[NasCode2026] Naslund, Eric. Unit-distance manuscript, numerical certificates and Lean formalization. Repository, with the corrected external certificates at commit 14289d9 (2 October 2026). The formalization status explains the remaining numerical hypothesis and the relation to the manuscript. The certificates of [Nas2026b], including the receipt for the hypothesis of [Pal2026b], are at commit b43f0c5 (10 October 2026).
[Pal2026] Naslund, Eric. An Exponent of 1.04273 for the Unit Distance Problem: Lean formalization at exponent 1.0427, conditional on one zeta inequality. Palomar registry, PALOMAR-2026-10-01-000018, version 1, registered 1 October 2026; formalization at commit e0ac836 of [NasCode2026].
[Nas2026b] Naslund, Eric. An Exponent of 1.043 for the Unit Distance Problem. Hexagon:2610.00016v2 (PDF), 10 October 2026. Theorem 1.1 and Remark 4.38.
[Pal2026b] Naslund, Eric. An Exponent of 1.043 for the Unit Distance Problem: Lean formalization at exponent 1.04315, conditional on one zeta inequality. Palomar registry, PALOMAR-2026-10-01-000018, version 2, registered 10 October 2026; formalization at commit edd88c0 of [NasCode2026].

Contribution notes

Prepared with assistance from Claude Opus 4.7, which retrieved abstracts of the May 2026 preprints and the OpenAI announcement. All references should be independently verified before citation.

The October 2026 update was prepared with assistance from OpenAI Codex. The $1.04317$ update and the upper bound $4/3 - \varepsilon$ were prepared with assistance from Claude Opus 5.5.

← Back to the table of constants