Erdős unit-distance conjecture

About 80 years old · traced to 1946

Let $N$ points be given in the Euclidean plane, and let a unit distance mean a pair of points whose Euclidean distance is $1$ . The unit distance conjecture. The number of unit distances determined by any $N$ points in the plane is always

\lesssim_\epsilon N^{1+\epsilon}

for every $\epsilon>0$ . This is an incidence-geometric conjecture related to replacing lines by circles in point-line incidence estimates; the source states it as an open conjecture and gives no resolution.

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Erdős unit-distance conjecture
If u(n) is the maximum number of unit-distance pairs among n planar points, is u(n) = n^(1+o(1))?
The unit distance conjecture in the plane
Let $\mathcal P$ be a set of $N$ points in the Euclidean plane, and let a unit distance be a pair of points in $\mathcal P$ whose Euclidean distance is $1$ . Unit distance conjecture. The number of unit distances determined by $N$ points in the plane is always
$\lesssim_\epsilon N^{1+\epsilon}$
for every $\epsilon>0$ . This is introduced in the context of incidence bounds, where analogous estimates for point–circle incidences are discussed; the source gives no resolution of the conjecture.
source: Ciprian Demeter, “Incidence theory and restriction estimates”, arXiv:1401.1873 (2014).

References

Primary source

Jean Bourgain and Ciprian Demeter, “The proof of the l^2 Decoupling Conjecture”, arXiv:1403.5335 (2015).

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 construction shows that planar point sets can have substantially more unit-distance pairs than the conjecture allowed, so the conjecture is false.

Paul Erdős posed the problem in 1946: whether the maximum number of unit-distance pairs among $n$ planar points satisfies $u(n)=n^{1+o(1)}$ .

Known results

Erdős’s constructions gave the lower bound $n^{1+\Omega(1/\log\log n)}$ .
Spencer, Szemerédi, and Trotter (1984) proved the general upper bound $O(n^{4/3})$ .

May 2026 disproof

An internal OpenAI model produced an infinite family with at least $n^{1+\varepsilon}$ unit distances for some fixed $\varepsilon>0$ , directly contradicting the conjecture. A companion paper gives a human-digested and externally verified proof; later work reports arbitrarily large examples exceeding $n^{1.0152}$ .

Current status (as of July 2026): The conjecture is disproved by a corroborated companion paper, while quantitative improvements to the exponent continue.

internal model at OpenAIOpenAIsolved2026-05-01evidence

Sources

No solutions have been posted yet.