On cycles in the coprime graph of integers (Q1378541)

Summary: The authors study cycles in the coprime graph of integers. Let \(f(n,k)\) denote the number of positive integers \(m\leq n\) with a prime factor among the first \(k\) primes. They show that there exists a constant \(c\) such that if \(A\subset \{1, 2, \ldots , n\}\) with \(|A|> f(n,2)\) (if \(6|n\) then \(f(n,2)=\frac{2}{3} n\)), 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 \(l\leq c n\).