{"docId":126,"paperId":126,"url":"https:\/\/hrj.episciences.org\/126","doi":"10.46298\/hrj.1993.126","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":27,"name":"Volume 16 - 1993"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01108688","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01108688v1","dateSubmitted":"2015-03-03 16:13:48","dateAccepted":"2015-06-12 16:05:34","datePublished":"1993-01-01 08:00:00","titles":{"en":"On sets of coprime integers in intervals"},"authors":["Erd\u00f6s, Paul","S\u00e1rk\u00f6zy, Andr\u00e1s"],"abstracts":{"en":"If $\\mathcal{A}\\subset\\mathbb{N}$ is such that it does not contain a subset $S$ consisting of $k$ pairwise coprime integers, then we say that $\\mathcal{A}$ has the property $P_k$. Let $\\Gamma_k$ denote the family of those subsets of $\\mathbb{N}$ which have the property $P_k$. If $F_k(n)=\\max_{\\mathcal{A}\\subset\\{1,2,3,\\ldots,n\\},\\mathcal{A}\\in\\Gamma_k}\\vert\\mathcal{A}\\vert$ and $\\Psi_k(n)$ is the number of integers $u\\in\\{1,2,3,\\ldots,n\\}$ which are multiples of at least one of the first $k$ primes, it was conjectured that $F_k(n)=\\Psi_{k-1}(n)$ for all $k\\geq2$. In this paper, we give several partial answers."},"keywords":[{"en":" prime number theorem"},{"en":"pairwise coprime integers"},"[MATH] Mathematics [math]"]}