On critical perfect systems of difference sets

AbstractA 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