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Erdős ProblemsCombinatorics

Erdős Problem 789: Cube Root Linearithmic

Let h(n)h(n) be maximal such that if A⊆ZA\subseteq \mathbb{Z} with ∣A∣=n\lvert A\rvert=n then there is B⊆AB\subseteq A with ∣B∣≥h(n)\lvert B\rvert \geq h(n) such that if a1+⋯+ar=b1+⋯+bsa_1+\cdots+a_r=b_1+\cdots+b_s with ai,bi∈Ba_i,b_i\in B then r=sr=s.

CombinatoricsAMS 5

Mathematical statement

Let h(n)h(n) be maximal such that if A⊆ZA\subseteq \mathbb{Z} with ∣A∣=n\lvert A\rvert=n then there is B⊆AB\subseteq A with ∣B∣≥h(n)\lvert B\rvert \geq h(n) such that if a1+⋯+ar=b1+⋯+bsa_1+\cdots+a_r=b_1+\cdots+b_s with ai,bi∈Ba_i,b_i\in B then r=sr=s.

Is h(n)=Θ((nlog⁡(n)))1/3)h(n) = \Theta((n\log(n)))^{1/3})?

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_789.variants.cube_root_linearithmic

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_789.variants.cube_root_linearithmic :    (fun n ↦ (subsetSumThreshold n : ℝ)) =Θ[atTop]      fun n ↦ (n * Real.log n) ^ ((1 : ℝ) / 3) := 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/789background