Complete tripartite subgraphs in the coprime graph of integers Original Research Article

We denote by f(n,k) the number of positive integers m ⩽ n with a prime factor among the first k primes. In this paper answering a question of Paul Erdős we show that there exist constants c, n0 such that if n ⩾ n0, A ⊂ {1,2,…,n} with |A| > f(n,2) (if 6|n then f(n,2) = 23n), then the coprime graph induced by A contains a complete tripartite graph on 2⌊clog nlog log log n⌋ + 1 vertices where one of the classes is a simple vertex and the other two classes each have ⌊clog nlog log log n⌋ vertices.