Let m,n1,n2,…,nm and c be positive integers. Let A = {A1, A2, … Am} be a system of sequences of integers Ai = (ai1 < a12 < … < aini; i=1,…,m, and let Di = {aij − aik ‖ 1 ⩽ k < j ⩽ ni} be the difference set of Aj. Then S = {D2, D2,…,Dm} is a perfect system of difference sets if Such a system is trivial if ni = 2 for at least one i. Paul Erdös conjectured that, for every positive integer e, except for a finite number of them, there is a non-trivial perfect system of difference sets whose differences are the first e positive integers. These exceptions are discussed and a proof of the conjecture is given.