On a problem of Erdős, Herzog and Schönheim

Let p1,p2,…,pnp1,p2,…,pn be distinct primes. In 1970, Erdős, Herzog and Schönheim proved that if DD, |D|=m|D|=m, is a set of divisors of N=p1α1⋯pnαn, α1≥α2≥⋯≥αnα1≥α2≥⋯≥αn, no two members of the set being coprime and if no additional member may be included in DD without contradicting this requirement then m≥αn∏i=1n−1(αi+1). They asked to determine all sets DD such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research.