Erdős Problem #959

Let A⊆ ℝ^2 be a set of size n and let d_1,…,d_k be the set of distinct distances determined by A. Let f(d) be the number of times the distance d is determined, ordered so that f(d_1)≥ f(d_2)≥ ⋯ ≥ f(d_k).

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-959,
  title        = {Erdős Problem #959},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-959}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $A \subseteq R^{2}$ be a set of size $n$ and let ${d_{1}, \dots, d_{k}}$ be the set of distinct distances determined by $A$ . Let $f (d)$ be the number of times the distance $d$ is determined, ordered so that $f (d_{1}) \geq f (d_{2}) \geq \dots \geq f (d_{k})$ . Estimate $max (f (d_{1}) - f (d_{2})),$ where the maximum is taken over all $A$ of size $n$ (this is extremalGap n).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«959». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_959 :
    (fun n ↦ (extremalGap n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → 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/959. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/959

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.