On a Problem of Erdős, Herzog and Schönheim

Abstract

Let $p_1, p_2,..., p_n$ be distinct primes. In 1970, Erd\H os, Herzog and Schönheim proved that if $\cal D$ is a set of divisors of $N=p_1^{α_1}...p_n^{α_n}$, $α_1\ge α_2\ge...\ge α_n$, 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 α_n \prod_{i=1}^{n-1} (α_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.