Erdős Problem #982

If n distinct points in ℝ^2 form a convex polygon then some vertex has at least lfloorn/2⌋ different distances to other vertices.

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-982,
  title        = {Erdős Problem #982},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-982}},
  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

If $n$ distinct points in $R^{2}$ form a convex polygon then some vertex has at least $⌊ \frac{n}{2} ⌋$ different distances to other vertices.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«982».

theorem erdos_982 (n : ℕ) (hn : 3 ≤ n) (p : Fin n → ℝ²) (hp : Function.Injective p)
    (hp' : EuclideanGeometry.IsConvexPolygon p) :
    ∃ (i : Fin n), { d : ℝ | ∃ j : Fin n, j ≠ i ∧ d = dist (p i) (p j) }.ncard ≥ n / 2

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/982. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/982

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.