An Existence Theory for Incomplete Designs

An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a 'hole'. If there are v points, a hole of size w, and all (other) block sizes equal k, this is denoted IPBD ((v; w), k). In addition to congruence restrictions on v and w, there is also a necessary inequality: v > (k - 1)w. This article establishes two main existence results for IPBD ((v; w), k): one in which w is fixed and v is large, and the other in the case v > (k - 1 + epsilon)w when w is large (depending on epsilon). Several possible generalizations of the problem are also discussed.