On a Problem of Erd\H{o}s, Herzog and Sch\"onheim

TL;DR

This paper solves a problem posed by Erdős, Herzog, and Schönheim regarding the characterization of divisor sets with specific coprimality constraints, providing a complete solution and proposing open questions.

Contribution

The paper fully characterizes the divisor sets that attain the minimal size bound under the given coprimality conditions, advancing understanding of divisor set structures.

Findings

Characterization of divisor sets meeting the minimal size bound.

Proof of the structure of extremal divisor sets.

Presentation of open problems for future research.

Abstract

Let $p_1, p_2,..., p_n$ be distinct primes. In 1970, Erd\H os, Herzog and Sch\"{o}nheim proved that if $\cal D$ is a set of divisors of $N=p_1^{\alpha_1}...p_n^{\alpha_n}$, $\alpha_1\ge \alpha_2\ge...\ge \alpha_n$, no two members of the set being coprime and if no additional member may be included in $\cal D$ without contradicting this requirement then $ |{\cal D}|\ge \alpha_n \prod_{i=1}^{n-1} (\alpha_i +1)$. They asked to determine all sets $\cal D$ such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research.