On cycles in the coprime graph of integers

Abstract

In this paper we study cycles in the coprime graph of integers. We denote by f(n, k) the number of positive integers m ≤ n with a prime factor among the first k primes. We show that there exists a constant c such that if A ⊂ {1,2,..., n} with |A| > f(n, 2) (if 6|n then f(n, 2) = 2/3n), then the coprime graph induced by A not only contains a triangle, but also a cycle of length 21 + 1 for every positive integer l ≤ cn.