Erdős Problem #786

The question

erdos_786.parts.i. Let $ϵ>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-ϵ$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\in A$ can only hold when $r=s$?

erdos_786.parts.ii. Is there some set $A\subset \{1,...,N\}$ of size $\ge (1-o(1))N$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\in A$ can only hold when $r=s$?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«786» (2 statements).

theorem erdos_786.parts.i : ∀ ε > 0, ε ≤ 1 →
    ∃ (A : Set ℕ) (δ : ℝ), 0 ∉ A ∧ 1 - ε < δ ∧ A.HasDensity δ ∧ A.IsMulCardSet

theorem erdos_786.parts.ii :
    ∃ (A : ℕ → Set ℕ) (f : ℕ → ℝ) (_ : f =o[atTop] (1 : ℕ → ℝ)),
    ∀ N, A N ⊆ Set.Icc 1 (N + 1) ∧ (1 - f N) * N ≤ (A N).ncard ∧ (A N).IsMulCardSet

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/786. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/786

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.