On 1-Sarvate–Beam designs

The solution to a set theory exercise, “Partition the set of positive integers { 1 , 2 , … , v } into k subsets such that the sum of the elements in each subset is v ( v + 1 ) / ( 2 k ) whenever v ( v + 1 ) / ( 2 k ) is an integer”, gives a construction of non-simple 1-SB designs. This raises a natural question of the existence of simple 1-SB designs. We show that the necessary conditions for the existence of simple 1-SB designs for block sizes 2, 3, 4, 5 and 6 are sufficient. Moreover, the technique exhibited in the proof can be applied to block sizes greater than k = 6 . We also show that simple t - SB ( v , t + 1 ) , 2 - SB ( v , 3 ) and 2 - SB ( v , 4 ) designs do not exist for any positive integers v and t . A natural question, “Can we obtain a construction for simple 1-SB designs similar to Billington’s classical construction of simple 1-designs for any block size k ?”, remains open.