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{\bf  On cycles in the coprime graph of integers \bigskip\noindent
Paul Erd\H{o}s
Gabor N. Sarkozy}
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In this paper we study cycles in the coprime graph of integers. 
We denote by $f(n,k)$ the number of positive integers $m\leq n$ with
a prime factor among the first $k$ primes.
 (If $6|n,$ then $f(n,2)={{2n}\over {3}} $.) We
show that there exists a constant $c$ such that if $A\subset \{1, 2,
\ldots , n\}$ with $|A| > f(n,2),$
then the coprime graph
induced by $A$ not only contains a triangle, but also a cycle of length 
$2 l + 1$ for every positive integer $l\leq c n .$
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