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 $\text{IPBD}\left( \left( v;w \right), \, k \right)$ . In addition to congruence restrictions on $v$ and $w$, there is also a necessary inequality: $v\, >\, \left( k\, -\, 1 \right)w$ . This article establishes two main existence results for $\text{IPBD}\left( \left( v;w \right), \, k \right)$ : one in which $w$ is fixed and $v$ is large, and the other in the case $v>\, \left( k-1+ε \right)w$ when $w$ is large (depending on $ε$ ). Several possible generalizations of the problemare also discussed.