Erdős problem 12 · problems.science

Let $A \subset \mathbb{N}$ be infinite with no distinct $a, b, c \in A$ such that $a \mid (b + c)$ with $b, c > a$ . Can $|A \cap [1, N]|/\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?

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10 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

12.lean

Retained formal statement5 of 9

Given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ , such that there are infinitely many $N$ such that $$\lvert A\cap\{1,\ldots,N\}\rvert > \frac{N}{f(N)}.

FormalConjectures/ErdosProblems/12.leanErdos12.erdos_12.variants.erdos_sarkozy3 linesExact file

1∀ (f : ℕ → ℕ),2  Filter.Tendsto f Filter.atTop Filter.atTop →3    ∃ A, Erdos12.IsGood A ∧ {N | ↑N / ↑(f N) < ↑(A ∩ Set.Icc 1 N).ncard}.Infinite

SolvedStatement only, no proof

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