Erdős Problem #104

Given n points in ℝ^2 the number of distinct unit circles containing at least three points is o(n^2).

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-104,
  title        = {Erdős Problem #104},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-104}},
  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

Given $n$ points in $R^{2}$ the number of distinct unit circles containing at least three points is $o (n^{2})$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«104».

theorem erdos_104 :
    (fun n : ℕ => (maxUnitCircleCount n : ℝ)) =o[atTop] (fun n : ℕ => (n : ℝ) ^ 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/104. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/104
[El84] Elekes, G., { $n$ } points in the plane can determine $n^{3/2}$ unit circles. Combinatorica (1984), 131.
[Er75h] Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.
[Er81d] Erdős, P., *Some applications of graph theory and combinatorial methods to number theory and geometry*. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
[HaMe86] Harborth, Heiko and Mengersen, Ingrid, Point sets with many unit circles. Discrete Math. (1986), 193--197.

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.