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Erdős ProblemsCombinatorics

Erdős Problem 488

Let AA be a finite set and B={n≥1:a∣n for some a∈A}.B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}. Is it true that, for every m>n≥max⁡(A)m>n\geq \max(A), ∣B∩[1,m]∣m<2∣B∩[1,n]∣n?\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?

CombinatoricsAMS 5AMS 11

Mathematical statement

Let AA be a finite set and B={n≥1:a∣n for some a∈A}.B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}. Is it true that, for every m>n≥max⁡(A)m>n\geq \max(A), ∣B∩[1,m]∣m<2∣B∩[1,n]∣n?\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_488

Canonical source
Complete statement target, proof intentionally absentLean 4
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 := by  sorry
Statement source
Formal Conjectures
Lean version
v4.27.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References

Pinned Formal Conjectures sourceformalizationerdosproblems.com/488background