Erdős Problem #488 — open problem in combinatorics

Erdős Problem #488

Let A be a finite set and B= n ≥ 1 : a| ntextrm for some a∈ A. Is it true that, for every m>n≥ max(A), lvert B∩ [1,m]rvert /m< 2lvert B∩ [1,n]rvert/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-488,
  title        = {Erdős Problem #488},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-488}},
  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

Let $A$ be a finite set and $B = {n \geq 1 : a ∣ n for some a \in A} .$ Is it true that, for every $m > n \geq max (A)$ , $\frac{∣ B \cap [ 1 , m ]∣}{m} < 2 \frac{∣ B \cap [ 1 , n ]∣}{n} ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«488». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_488 : answer(sorry) ↔ ∀ (A : Finset ℕ), A.Nonempty →
    -- These are needed for the reasons outlined here: https://github.com/google-deepmind/formal-conjectures/pull/256
    0 ∉ A → 1 ∉ A →
    letI B := {n ≥ 1 | ∃ a ∈ A, a ∣ n}
    ∀ᵉ (n : ℕ) (m > n), A.max ≤ n →
      ((Finset.Icc 1 m).filter (· ∈ B)).card / (m : ℚ) <
        2 * ((Finset.Icc 1 n).filter (· ∈ B)).card / 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/488. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/488

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.