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Level A · Machine-checkable Hard Geometry P-erdos-100

Erdős Problem #100

Is the diameter of A at least Cn for some constant C > 0?

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-100,
  title        = {Erdős Problem #100},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-100}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Verified
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Disputed
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Refuted
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On the literature board
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Current state

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

The problem

The question

Is the diameter of at least for some constant ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«100». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_100 :
    answer(sorry) ↔ ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ A : Finset ℝ²,
      A.card = n →
      DistancesSeparated A →
      diam (A : Set ℝ²) > C * 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/100. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_100.variants.strong — Stronger conjecture: diameter ≥ n - 1 for sufficiently large n.

References

  • erdosproblems.com/100
  • [Kanold](No references found)
  • [GuKa15](Guth, Larry and Katz, Nets Hawk, On the Erd\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.)
  • [Piepmeyer](No references found)

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.