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 m 0 = m 0 ( k , u ) such that there exists a group divisible 3-design of group type m u with block size k and index one for all integers m ⩾ m 0 if and only if1. ≡ 0 ( mod ( k − 2 ) ) , 1 ) ( u − 2 ) ≡ 0 ( mod ( k − 1 ) ( k − 2 ) ) , − 1 ) ( u − 2 ) ≡ 0 ( mod k ( k − 1 ) ( k − 2 ) ) . logous 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of this theorem gives an asymptotic existence result for transversal 3-designs which was proved by Blanchard in his unpublished manuscript as well.