Let $\epsilon>0$ and $N$ be sufficiently large. If $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert \geq \epsilon N$ then must there exist $a_1,a_2,a_3\in A$ and distinct primes $p_1,p_2,p_3$ such that $$a_1p_1=a_2p_2=a_3p_3?$$

1False ↔2  ∀ (ε : ℝ),3    0 < ε →4      ∀ᶠ (N : ℕ) in Filter.atTop,5        ∀ A ⊆ Finset.Icc 1 N,6          ↑A.card ≥ ε * ↑N →7            ∃ a₁ ∈ A,8              ∃ a₂ ∈ A,9                ∃ a₃ ∈ A,10                  ∃ p₁ p₂ p₃,11                    Nat.Prime p₁ ∧12                      Nat.Prime p₂ ∧ Nat.Prime p₃ ∧ p₁ ≠ p₂ ∧ p₁ ≠ p₃ ∧ p₂ ≠ p₃ ∧ a₁ * p₁ = a₂ * p₂ ∧ a₂ * p₂ = a₃ * p₃