Pairwise balanced designs of dimension three

The dimension of a pairwise balanced design PBD(v,K) is the maximum positive integer d such that any d of its points are contained in a proper flat. The standard examples are affine and projective linear spaces. Also, Teirlinck provided a nearly complete existence theory for ‘Steiner spaces’, which is the case K = {3} and d = 3. A recent result of the first author and A.C.H. Ling says that there exists a PBD(v,K) of dimension at least d for all sufficiently large and numerically admissible v. Here, we consider the detailed existence question for K = {3, 4, 5} and d = 3. A certain problem on latin squares connects with this particular case.