Math Conjectures

Exact Growth of the Planar Unit-Distance Function

OPENMajorOpen problemProposed 1946 · Standard version

Canonical statement

For n≥2n\geq2, set
u(n)=max⁡P⊂R2∣P∣=n∣{ {x,y}⊂P:∥x−y∥2=1 }∣ u(n)=\max_{\substack{P\subset\mathbb R^2\\|P|=n}} \bigl|\{\,\{x,y\}\subset P:\|x-y\|_2=1\,\}\bigr|
and
αU=lim sup⁡n→∞log⁡u(n)log⁡n. \alpha_U=\limsup_{n\to\infty}\frac{\log u(n)}{\log n}.
Determine the exact value of αU\alpha_U (and, more finely, the asymptotic order of u(n)u(n)).
View source LaTeX
For \(n\geq2\), set
\[
  u(n)=\max_{\substack{P\subset\mathbb R^2\\|P|=n}}
    \bigl|\{\,\{x,y\}\subset P:\|x-y\|_2=1\,\}\bigr|
\]
and
\[
  \alpha_U=\limsup_{n\to\infty}\frac{\log u(n)}{\log n}.
\]
Determine the exact value of \(\alpha_U\) (and, more finely, the
asymptotic order of \(u(n)\)).
DG-003:[SpencerSzemerediTrotter1984Unit][OpenAI2026UnitDistance][SawinEtAl2026UnitDistance]
Discrete geometryCombinatorics

Notes

This problem asks for the maximal number u(n)u(n) of unit-distance pairs among nn points in the plane — a question going back to Erdős in 1946 — and specifically for the growth exponent αU=lim sup⁡n→∞log⁡u(n)/log⁡n\alpha_U=\limsup_{n\to\infty}\log u(n)/\log n. Sections of a suitably scaled integer lattice show that u(n)u(n) grows slightly faster than linearly, and Erdős long conjectured that near-linear growth of the form n1+O(1/log⁡log⁡n)n^{1+O(1/\log\log n)} is the truth.

On the upper side, the incidence bound of Spencer, Szemerédi, and Trotter gives u(n)=O(n4/3)u(n)=O(n^{4/3}), hence αU≤4/3\alpha_U\le 4/3 [SpencerSzemerediTrotter1984Unit]. In May 2026 the near-linear conjecture was disproved: an externally checked construction, found with the assistance of an AI model, produces an absolute δ>0\delta>0 and infinitely many nn with u(n)≥n1+δu(n)\ge n^{1+\delta} [OpenAI2026UnitDistance] [SawinEtAl2026UnitDistance].

What survives is the exact-growth problem: αU\alpha_U is now pinned between 1+δ1+\delta and 4/34/3, and determining its exact value — and the finer asymptotic order of u(n)u(n) — remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.