# zbMATH — the first resource for mathematics

An existence theory for incomplete designs. (English) Zbl 1337.05014
Summary: 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 $$\operatorname{IPBD}((v;w),k)$$. In addition to congruence restrictions on $$v$$ and $$w$$, there is also a necessary inequality: $$v > (k-1)w$$. This article establishes two main existence results for $$\operatorname{IPBD}((v;w),k)$$: one in which $$w$$ is fixed and $$v$$ is large, and the other in the case $$v > (k-1+\epsilon) w$$ when $$w$$ is large (depending on $$\epsilon$$). Several possible generalizations of the problem are also discussed.

##### MSC:
 05B30 Other designs, configurations 05B05 Combinatorial aspects of block designs 05C65 Hypergraphs
##### Keywords:
block design; hypergraph block design; hypergraph
Full Text: