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Erdős ProblemsCombinatorics

Erdős Problem 535

Let r≥3r \geq 3, and let fr(N)f_r(N) denote the size of the largest subset of {1,…,N}\{1,\ldots,N\} such that no subset of size rr has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that f3(N)>Nc/log⁡log⁡Nf_3(N) > N^{c/\log\log N} for some cons...

CombinatoricsAMS 5AMS 11

Mathematical statement

Let r≥3r \geq 3, and let fr(N)f_r(N) denote the size of the largest subset of {1,…,N}\{1,\ldots,N\} such that no subset of size rr has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that f3(N)>Nc/log⁡log⁡Nf_3(N) > N^{c/\log\log N} for some constant c>0c > 0, and conjectured this should also be an upper bound; here we state the conjectural upper bound for all r≥3r \geq 3.

See also [536].

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_535

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_535 : ∀ r ≥ 3, ∃ c > (0 : ℝ),    ∀ᶠ (N : ℕ) in atTop,      (f r N : ℝ) ≤ (N : ℝ) ^ (c / log (log (N : ℝ))) := 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/535background