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# An Existence Theory for Incomplete Designs

Published:2016-02-01
Printed: Jun 2016
• Peter Dukes,
Department of Mathematics and Statistics, University of Victoria, Victoria, Canada
• E.R. Lamken,
773 Colby Street, San Francisco, CA, USA 94134
• Alan C.H. Ling,
Department of Computer Science, University of Vermont, Burlington, VT, USA 05405
 Format: LaTeX MathJax PDF

## 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: $v \gt (k-1)w$. 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 $v \gt (k-1+\epsilon) w$ when $w$ is large (depending on $\epsilon$). Several possible generalizations of the problem are also discussed.
 Keywords: block design, hypergraph
 MSC Classifications: 05C70 - Factorization, matching, partitioning, covering and packing

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