On a conjecture of Erdős about sets without $k$ pairwise coprime integers

Abstract

Let ${\mathbb{Z}}^{+}$ be the set of positive integers. Let $C_k$ denote all subsets of ${\mathbb{Z}}^{+}$ such that neither of them contains $k+1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\dots ,n\}$. Let $f(n,k)={\text{max}}_{A\in C_k(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_1,\dots ,p_k$, where $p_i$ denote the $i$th prime number. In 1962, Erdős conjectured that $f(n,k)=|E(n,k)|$ for every $n\ge p_k$. Recently Chen and Zhou proved some results about this conjecture. In this paper we solve an open problem of Chen and Zhou and prove several related results about the conjecture.

19 pages