An Existence Theorem for Group Divisible Designs of Large Order

The following result gives a partial answer to a question of R. M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers, 2⩽k⩽u. Then there exists an integer m0=m0(k, u) such that there exists a group divisible design of group type mu with block size k and index one for all integer m⩾m0 if and only if (i) u−1≡0 mod(k−1), (ii) u(u−1)≡0 mod k(k−1). This is a generalization of the well-known result of Chowla, Erdős, and Straus on the existence of transversal designs of large order.