An existence theorem for group divisible 3-designs of large order

In this paper we establish an asymptotic existence result for group divisible 3-designs of large order. Let k and u be positive integers, 3⩽k⩽u. Then there exists an integer m0=m0(k,u) such that there exists a group divisible 3-design of group type mu with block size k and index one for all integers m⩾m0 if and only if1.,2.,3..An analogous theorem was proved by Mohácsy and Ray-Chaudhuri for group divisible 2-designs in a previously published paper in 2002. The u=k case of this theorem gives an asymptotic existence result for transversal 3-designs which was proved by Blanchard in his unpublished manuscript as well.