The existence theorem for group divisible designs of large order (Q1601428)

The authors generalize a result of Chowla, Erdős, and Straus on the existence of transversal designs of large order. Let \(k,u\in\mathbb{Z}^+\) such that \(2\leq k\leq u\). The authors establish the existence of an integer \(m_0= m_0(k,u)\) such that a group divisible design of group type \(m^u\) with block size \(k\) and index \(1\) exists for all integers \(m\geq m_0\) if and only if \(u-1\equiv 0\text{\;mod} (k-1)\) and \(u(u- 1)\equiv 0\text{\;mod }k(k-1)\). This partially answers a question of R. M. Wilson concerning the existence of group divisible designs of larger order.