Elsevier

Discrete Mathematics

Volume 47, 1983, Pages 255-266
Discrete Mathematics

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

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Abstract

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 = {aijaik ‖ 1 ⩽ k < jni} be the difference set of Aj. Then S = {D2, D2,…,Dm} is a perfect system of difference sets if D=D1⋃D2⋃⋯⋃Dm=c,c+1,…,c−1+i=1mni2 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.

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