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An Existence Theory for Incomplete Designs

Published online by Cambridge University Press:  20 November 2018

Peter Dukes ,
Esther R. Lamken  and
Alan C. H. Ling
Peter Dukes
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC e-mail: dukes@uvic.ca
Esther R. Lamken
Affiliation:
66h Colby Street, San Francisco, CA, USA 94134 e-mail: lamken@caltech.edu
Alan C. H. Ling
Affiliation:
Department of Computer Science, University of Vermont, Burlington, VT, USA 05405 e-mail: aling@emba.uvm.edu
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Abstract

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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.

Keywords

05C70block designhypergraph

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 287 - 302
Copyright
Copyright © Canadian Mathematical Society 2016

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