Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Geometry P-erdos-1083

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)?

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.

Start working on it Submit a claim Follow
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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let , and let be the minimal such that every set of points in determines at least distinct distances. Estimate - in particular, is it true that

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1083». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1083 : answer(sorry) ↔
    ∀ 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 {} 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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.