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 if 1. u - 2 equivalent to 0 (mod (k - 2)), 2. (u - 1)(u - 2) equivalent to 0 (mod (k - 1)(k - 2)), 3. u(u - 1)(u - 2) equivalent to 0 (mod k(k - 1)(k - 2)). An analogous theorem was proved by Mohacsy 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of this theorem gives an asymptotic existence result for transversal 3-designs which was proved by Blanchard in his unpublished manuscript as well. (C) 2010 Elsevier Inc. All rights reserved.