Halving Steiner 2-designs

A Steiner 2-design S(2,k,v)S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G   on vv vertices into KksKks. The obvious necessary condition of those orders vv for which there exists a halvable S(2,k,v)S(2,k,v) is that vv admits the existence of an S(2,k,v)S(2,k,v) with an even number of blocks. In this paper, we give an asymptotic solution for various block sizes. We prove that for any k⩽5k⩽5 or any Mersenne prime k  , there is a constant number v0v0 such that if v>v0v>v0 and vv satisfies the above necessary condition, then there exists a halvable S(2,k,v)S(2,k,v). We also show that a halvable S(2,2n,v)S(2,2n,v) exists for over a half of possible orders. Some recursive constructions generating infinitely many new halvable Steiner 2-designs are also presented.