On 1-Sarvate–Beam designs

The solution to a set theory exercise, “Partition the set of positive integers {1,2,…,v}{1,2,…,v} into kk subsets such that the sum of the elements in each subset is v(v+1)/(2k)v(v+1)/(2k) whenever v(v+1)/(2k)v(v+1)/(2k) 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=6k=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 vv and tt.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 kk?”, remains open.