Erdős Problem #1088

Let f_d(n) be the minimal m such that any set of m points in ℝ^d contains a set of n points for which any two determined distances are distinct. Erdős Problem 1088 asks to estimate f_d(n). In particular, is it true that, for every fixed n ≥ 3, f_d(n) = 2^o(d) as d → ∞?

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-1088,
  title        = {Erdős Problem #1088},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1088}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $f_{d} (n)$ be the minimal $m$ such that any set of $m$ points in $R^{d}$ contains a set of $n$ points for which any two determined distances are distinct. Erdős Problem 1088 asks to estimate $f_{d} (n)$ . In particular, is it true that, for every fixed $n \geq 3$ , $f_{d} (n) = 2^{o (d)}$ as $d \to \infty$ ?

The little- $o$ condition is stated after taking the base- $2$ logarithm.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1088». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1088 :
    answer(sorry) ↔
      ∀ n ≥ 3,
        (fun d : ℕ ↦ Real.logb 2 (f d n : ℝ)) =o[atTop] (fun d : ℕ ↦ (d : ℝ))

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/1088. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1088

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.