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Let $f(n, k) = \\text{max}_{A \\in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \\dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erd\\H{o}s conjectured that $f(n, k) = |E(n,k)|$ for every $n \\ge p_{k}$. 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Kiss","submitted_at":"2017-05-16T14:39:07Z","abstract_excerpt":"Let $\\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\\cap \\{1,2,\\ldots,n\\}$. Let $f(n, k) = \\text{max}_{A \\in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \\dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erd\\H{o}s conjectured that $f(n, k) = |E(n,k)|$ for every $n \\ge p_{k}$. 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