Source labels openChecked July 26, 2026

Erdős Problem 539: Sq Cube Root

Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$ , the set $\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$ has size at least $h(n)$ . Is $h(n) = \Theta(n^{2/3})$ ?

Combinatorics AMS 5 AMS 11

Mathematical statement

Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$ , the set $\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$ has size at least $h(n)$ . Is $h(n) = \Theta(n^{2/3})$ ?

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_539.variants.sq_cube_root

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_539.variants.sq_cube_root :    (fun n ↦ (cofactorThreshold n : ℝ)) =Θ[atTop] fun n ↦ (n : ℝ) ^ ((2 : 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 sourceformalization erdosproblems.com/539background