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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 776.11011

**Autor: ** Erdös, Paul; Sárközy, A.

**Title: ** On sets of coprime integers in intervals. (In English)

**Source: ** Hardy-Ramanujan J. 16, 1-20 (1993).

**Review: ** For any subset *A* of **N**, let \Phi_{k}(*A*) denote the number of k-tuples (a_{1},...,a_{k}) with a_{i} in *A*, a_{1} < a_{2} < ··· < a_{k} and (a_{i},a_{j}) = 1 for 1 \leq i < j \leq k and let \Gamma_{k} denote the family of those *A* with \Phi_{k}(*A*) = 0. Further, define g_{k}(m,n) to be the maximal cardinality of a set *A* in \Gamma_{k} and lying in [m,m+n-1] and write F_{k}(n) = g_{k}(1,n) and G_{k}(n) = **max**_{m in N gk(m,n)}. Define also the functions \psi_{k}(m,n) and \Psi_{k}(n) by \psi_{k}(m,n) = |**{**u in **N**: u in [m,n+n-1], u divisible by at least one of the first k primes| and \Psi_{k}(n) = \psi_{k}(1,n). Finally, let h_{(k,l)}(m,n) denote the maximal cardinality of a set *A* in \Gamma_{l} and lying in [m,n+n-1] with the property that each a in *A* is not divisible by any of the first k primes.

The first author has conjectured that F_{k}(n) = \Psi_{k-1}(n) for each k and this has been confirmed for k \leq 4. In this paper, the authors study the case of general k and prove seven theorems concerning connections between the various functions mentioned above.

**Reviewer: ** M.Nair (Glasgow)

**Classif.: ** * 11B83 Special sequences of integers and polynomials

**Keywords: ** sets of coprime integers in intervals; maximal cardinality

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