Multiplicative bases and an Erdős problem

Abstract

In this paper we investigate how small the density of a multiplicative basis of order $h$ can be in $\{1,2,\dots ,n\}$ and in ${\mathbb{Z}}^{+}$. Furthermore, a related problem of Erd\H os is also studied: How dense can a set of integers be, if none of them divides the product of $h$ others?