Canadian Mathematical Society
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An Existence Theory for Incomplete Designs

 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
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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 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 block design, hypergraph
MSC Classifications: 05C70 show english descriptions Factorization, matching, partitioning, covering and packing 05C70 - Factorization, matching, partitioning, covering and packing

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