An Existence Theory for Incomplete Designs. Issue 2 (1st June 2016)

• Record Type: Journal Article
• Authors: Dukes, Peter; Lamken, Esther R.; Ling, Alan C. H.
• Is Part Of: Canadian mathematical bulletin =. Volume 59:Issue 2(2016)
• Page Start: 287
• Page End: 302
• Publication Date: 2016-06-01
• Subjects: 05C70; block design -- hypergraph; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2015-073-7
• Languages: English
• ISSNs: 0008-4395

Abstract

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 $v>\, \left( k-1+\varepsilon \right)w$ when $w$ is large (depending on $\varepsilon$ ). Several possible generalizations of the problemare also discussed.