Erdős problem 13 · problems.science

If $A \subseteq \{1, ..., N\}$ is a set with no $a, b, c \in A$ such that $a | (b+c)$ and $a < \min(b,c)$ , then $|A| \le N/3 + O(1)$ . This has been solved by Bedert [Be23].

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2 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

13.lean

Retained formal statement2 of 2

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 $|A| \le N/(r+1) + O(1)$ ?

FormalConjectures/ErdosProblems/13.leanErdos13.erdos_13.variants.general7 linesExact file

1True ↔2  ∀ (r : ℕ),3    ∃ C,4      ∀ (N : ℕ),5        ∀ A ⊆ Finset.Icc 1 N,6          (∀ a ∈ A, ∀ (b : Fin r → ℕ), (∀ (i : Fin r), b i ∈ A) → (∀ (i : Fin r), a < b i) → ¬a ∣ ∑ i, b i) →7            ↑A.card ≤ ↑N / (↑r + 1) + C

OpenStatement only, no proof

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