Pairwise balanced designs with holes

We consider the question of the existence of pairwise balanced designs on n + s points containing a block of cardinality s and all other blocks of cardinality at least 3. We denote such a design by PBDH(n + s:s). Building on earlier work of Rees, Mendelsohn, Jungnickel and Lenz, we show that necessary conditions for the existence of a PBDH(n + s:s) with n even are 1 ≤ s ≤ n- 1 and s n- 2 when n = 2 or 4 (mod 6), and 1 ≤ s ≤ n- 1 when n = 0 (mod 6). We show that these conditions are also sufficient except in the following instances, when such a design does not exist: (n, s) ε ((6,2),(8,2),(10,2)). In the case when n is odd a necessary condition is that 1 ≤ s ≤ n(n- 1)/(n + 3). This condition is proved to be sufficient for all n ≥ 37. For n ≤ 35 we show that a PBDH(n + s:s) does not exist when (n, s) ε ((5,2),(7,2),(7,4).(11,2)) and that there are at most 16 other possible exceptional pairs (n, s).