A general version asks, for a fixed $r \in \mathbb{N}$, if a set
$A \subseteq \{1, ..., N\}$ has no $a \in A$ and $b_1, ..., b_r \in A$ such that
$a | (b_1 + ... + b_r)$ and $a < \min(b_1, ..., b_r)$, then is it true that
$|A| \le N/(r+1) + O(1)$?
Actions
Submit a Proof
Have a proof attempt? Submit it for zero-trust verification.
theorem erdos_13_general : answer(sorry) ↔ ∀ r : ℕ, ∃ C : ℝ, ∀ N : ℕ,
∀ A ⊆ Icc 1 N,
(∀ a ∈ A, ∀ (b : Fin r → ℕ), (∀ i, b i ∈ A) → (∀ i, a < b i) →
¬ (a ∣ ∑ i, b i)) →
(A.card : ℝ) ≤ (N : ℝ) / (r + 1) + C
