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 $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. 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.