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Level A · Machine-checkable Hard Geometry P-erdos-98

Erdős Problem #98

Let h(n) be such that any n points in ℝ^2, with no three on a line and no four on a circle, determine at least h(n) distinct distances. Does h(n)/n→ ∞?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

  • erdosproblems.com lists Erdős problem #98 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement erdos_98 as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #98 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `erdos_98`research openFormal Conjectures
Formal Conjectures: research statements in the file1 open, 1 solvedFormal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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-98,
  title        = {Erdős Problem #98},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-98}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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On the literature board
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Nobody has worked on this problem here yet

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  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

or , ,

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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

Let be such that any points in , with no three on a line and no four on a circle, determine at least distinct distances. Does ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«98».

theorem erdos_98 :
    Tendsto (fun n : ℕ ↦ ((h n : ℝ) / (n : ℝ))) atTop atTop

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/98. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • [Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
  • [Er83c] Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.
  • [Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
  • [Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
  • [Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
  • [EFPR93] Erdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189-196.
  • [Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.
  • [Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
  • erdosproblems.com/98

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.