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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
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