Halving the Complete Design

Let X be a finite set of cardinality v. We denote the set of all k-subsets of X by In this paper we consider the problem of partitioning into two parts of equal size, each of which is the block set of a 2-(v k, λ) design. We determine necessary and sufficient conditions on v for the existence of such a partition when k=3 or 4. We also construct partitions for higher values of fc and infinitely many values of v. The case where k=3 has been solved for all values of v by Dehon [5]. The case where v=2k has been solved for all values of k by Alltop [1], The remaining results are new. The technique used is similar to that used by Denniston in his construction of a 4-(12,5,4) design without repeated blocks. We also prove an interesting corollary to Baranyai's theorem giving necessary and sufficient conditions for the existence of a partition of into 1- (v, k,λ) designs.