OpenFrontiers
Join
← Back to the index
OP-9338OpenGeometry and topology

Erdős #1085 · geometry / distances

Let fd(n)f_d(n) be minimal such that, in any set of nn points in Rd\mathbb{R}^d, there exist at most fd(n)f_d(n) pairs of points which distance 11 apart. Estimate fd(n)f_d(n).

Source reports OPENAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let fd(n)f_d(n) be minimal such that, in any set of nn points in Rd\mathbb{R}^d, there exist at most fd(n)f_d(n) pairs of points which distance 11 apart. Estimate fd(n)f_d(n).

Lean 4

No Lean submitted

Formal statement

The formalization is available at the source below; its code is not attached here yet.

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.

Proofs & verification

No Lean has been submitted for this problem yet. A Lean file can be attached to a submission for review, though with no target here it cannot be machine-checked.

Share a formalization →Verification guide ↗

Background

The most difficult cases are d=2d=2 and d=3d=3. When d=2d=2 this is the unit distance problem [90], and the best known bounds are

n1+c<f2(n)≪n4/3n^{1+c}< f_2(n) \ll n^{4/3}

for some small constant c>0c>0, the lower bound by an internal model at OpenAI, at least for infinitely many nn (see [90]) and the upper bound by Spencer, Szemerédi, and Trotter [SST84].

When d=3d=3 the best known bounds are

n4/3log⁡log⁡n≪f3(n)≪n3/2β(n)n^{4/3}\log\log n \ll f_3(n) \ll n^{3/2}\beta(n)

where β(n)\beta(n) is a very slowly growing function, the lower bound by Erdős [Er60b] and the upper bound by Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl [CEGSW90].

A construction of Lenz (taking points on orthogonal circles) shows that, for d≥4d\geq 4,

fd(n)≥p−12pn2−O(1)f_d(n)\geq \frac{p-1}{2p}n^2-O(1)

with p=⌊d/2⌋p=\lfloor d/2\rfloor. Erdős [Er60b] showed that the Erdős-Stone theorem implies

fd(n)≤(p−12p+o(1))n2f_d(n) \leq \left(\frac{p-1}{2p}+o(1)\right)n^2

for d≥4d\geq 4.

Erdős [Er67e] determined fd(n)f_d(n) up to O(1)O(1) for all even d≥4d\geq 4. Brass [Br97] determined f4(n)f_4(n) exactly. Swanepoel [Sw09] determined fd(n)f_d(n) exactly for even d≥6d\geq 6. For odd d≥5d\geq 5 Erdős and Pach [ErPa90] proved that there exist constants c1(d),c2(d)>0c_1(d),c_2(d)>0 such that

p−12pn2+c1n4/3≤fd(n)≤p−12pn2+c2n4/3.\frac{p-1}{2p}n^2 +c_1n^{4/3}\leq f_d(n) \leq \frac{p-1}{2p}n^2 +c_2n^{4/3}.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1085 link
  2. [2]Upstream Lean 4 statement for Erdős #1085 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1085 link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.