  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.
