On critical perfect systems of difference sets Original Research Article

A perfect system of difference sets with threshold c is a partition of a consecutive run of integers beginning with c into full difference sets of valency at least 2. The BKT inequality, due to Bermond, Kotzig and Turgeon gives a necessary condition for the existence of such systems; systems for which this inequality holds with equality are called critical. We show that a critical perfect system of difference sets with threshold c which contains no difference sets of valency 2 consists of 2c - 1 difference sets of valency 3. We also discuss, in the light of this result, other inequalities for perfect systems which are stronger than the BKT inequality at least in some circumstances.