Erdős Problem #653 — open problem in combinatorics

Erdős Problem #653

Let x_1,…,x_n∈ ℝ^2 and let R(x_i)=\# lvert x_j-x_irvert : j≠ i, where the points are ordered such that R(x_1)≤ ⋯ ≤ R(x_n). Let g(n) be the maximum number of distinct values the R(x_i) can take. Is it true that g(n) ≥ (1-o(1))n?

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-653,
  title        = {Erdős Problem #653},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-653}},
  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 $x_{1}, \dots, x_{n} \in R^{2}$ and let $R (x_{i}) = # {∣ x_{j} - x_{i} ∣ : j \neq = i}$ , where the points are ordered such that $R (x_{1}) \leq \dots \leq R (x_{n}) .$ Let $g (n)$ be the maximum number of distinct values the $R (x_{i})$ can take. Is it true that $g (n) \geq (1 - o (1)) n$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«653». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_653 : answer(sorry) ↔ ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧
    ∀ᶠ n in atTop, (1 - o n) * n ≤ maximalDistinctDistancesFrom 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/653. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/653

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.