Multiplicative bases and an Erd\H{o}s problem

TL;DR

This paper explores the minimal density of multiplicative bases of a given order within finite sets and positive integers, and examines Erdős's related problem on the density of sets avoiding divisibility of products of others.

Contribution

It introduces new bounds and insights into the density of multiplicative bases and sets avoiding certain divisibility conditions, extending classical problems in number theory.

Findings

Derived bounds for the density of multiplicative bases of order h.

Established limitations on the density of sets avoiding divisibility of products.

Connected the problems to Erdős's conjectures and provided new perspectives.

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?