Erdős (Erdos) Problem #1083: status in 2026

Erdős Problem #1083

Let d≥ 3, and let f_d(n) be the minimal m such that every set of n points in ℝ^d determines at least m distinct distances. Estimate f_d(n) - in particular, is it true that f_d(n)=n^2/d-o(1)?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-10-17).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1083 as open (status updated 17 October 2025).
Formal Conjectures marks the Lean statement erdos_1083 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1083 on erdosproblems.com	open 2025-10-17	erdosproblems.com
Formal Conjectures: status of `erdos_1083`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-1083,
  title        = {Erdős Problem #1083},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1083}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Let $d \geq 3$ , and let $f_{d} (n)$ be the minimal $m$ such that every set of $n$ points in $R^{d}$ determines at least $m$ distinct distances. Estimate $f_{d} (n)$ - in particular, is it true that $f_{d} (n) = n^{\frac{2}{d} - o (1)} ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1083».

theorem erdos_1083 :
    ∀ d : ℕ, 3 ≤ d → ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧
      ∀ᶠ n : ℕ in atTop,
        (minimalDistinctDistances (ℝ^d) n : ℝ) = (n : ℝ) ^ ((2 : ℝ) / (d : ℝ) - o n)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1083. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1083
[APST04] Aronov, Boris and Pach, János and Sharir, Micha and Tardos, Gábor, *Distinct distances in three and higher dimensions*. Combin. Probab. Comput. (2004), 283--293.
[CEGSW90] Clarkson, Kenneth L. and Edelsbrunner, Herbert and Guibas, Leonidas J. and Sharir, Micha and Welzl, Emo, Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom. (1990), 99--160.
[Er46b] Erdős, P., On sets of distances of { $n$ } points. Amer. Math. Monthly (1946), 248--250.
[SoVu08] Solymosi, József and Vu, Van H., *Near optimal bounds for the {E}rdős distinct distances problem in high dimensions*. Combinatorica (2008), 113--125.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.