On a conjecture of Paul Erdös for perfect systems of difference sets

Abstract

Let m, n 1, n 2,…, n m and c be positive integers. Let A = { A 1, A 2, … A m} be a system of sequences of integers A i = ( a i1 < a 12 < … < a in i ; i=1,…, m, and let D i = { a ij − a ik ‖ 1 ⩽ k < j ⩽ n i } be the difference set of A j . Then S = { D 2, D 2,…, D m} is a perfect system of difference sets if D=D 1⋃D 2⋃⋯⋃D m= c,c+1,…,c−1+ ∑ i=1 m n i 2 Such a system is trivial if n i = 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.