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Erdős ProblemsNumber theory

Erdős Problem 536

Let ϵ>0\epsilon>0 and NN be sufficiently large. Is it true that if A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has size at least ϵN\epsilon N then there must be distinct a,b,c∈Aa,b,c\in A such that [a,b]=[b,c]=[a,c],[a, b]=[b, c]=[a, c], where [⋅,⋅][\cdot, \cdot] denotes the least common multiple?

Number theoryAMS 11

Mathematical statement

Let ϵ>0\epsilon>0 and NN be sufficiently large. Is it true that if A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has size at least ϵN\epsilon N then there must be distinct a,b,c∈Aa,b,c\in A such that [a,b]=[b,c]=[a,c],[a, b]=[b, c]=[a, c], where [⋅,⋅][\cdot, \cdot] denotes the least common multiple?

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_536

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_536 :    answer(sorry) ↔ ∀ᵉ (ε > (0: ℝ)), ∀ᶠ N in atTop,    ∀ (A : Finset ℕ), A ⊆ Icc 1 N → (ε * (N : ℝ)) ≤ (A.card : ℝ) →    ∃ᵉ  (a ∈ A) (b ∈ A) (c ∈ A),    # {a, b, c} = 3 ∧ a.lcm b = b.lcm c ∧ b.lcm c = a.lcm c := 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/536background