An Existence Theory for Incomplete Designs
Printed: Jun 2016
Alan C.H. Ling,
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.
block design, hypergraph
05C70 - Factorization, matching, partitioning, covering and packing