An Existence Theory for Incomplete Designs. Issue 2 (1st June 2016) - Dukes, Peter; Lamken, Esther R.; Ling, Alan C. H.
Abstract: 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+ε )w when w is large (depending on ε ). Several possible generalizations of the problemare also discussed.