Proportionally balanced designs: Some further results and general constructions (Q2725023)

A balanced block design \(S(\lambda; 2,k,v)\) is a pair \((S,\beta)\), where \(S\) is a \(v\)-set and \(\beta\) is a collection of \(k\)-subsets of \(S\), called blocks, such that any 2-subset of \(S\) is contained in exactly \(\lambda\) blocks. The most often used generalization of this notation to multiple block sizes is to replace the positive integer \(k\geq 2\) by a set \(K\) of allowed block sizes. Proportionally balanced designs provide an alternative generalization. A proportionally balanced design is a pair \((S,\beta)\), where \(S= \{p_1,\dots, p_n\}\) is a set, \(v> 3\), and \(\beta\) consists of a collection of subsets of \(S\), \(|B|\geq 2\) for all \(B\in \beta\), such that, if \(r_i\) and \(p_{ij}\), \(i\neq j\), denote the number of blocks through \(p_i\) (\(p_i\) and \(p_j\), respectively), then \(\lambda_{ij}> 0\) for all \(i\neq j\) and \(r_i/r_j= \lambda_{im}/\lambda_{jm}\) for all pairwise distinct \(i\), \(j\), \(m\). Proportionally balanced designs were introduced by \textit{K. Gray} and \textit{G. Matters} [Util. Math. 48, 33-64 (1995; Zbl 0875.62588)] in response to a need for the allocation of markers of the Queensland core skills test to have a certain property. They proved that if all blocks have the same size, then a design is proportionally balanced iff it is a balanced block design. In the present paper, further theoretical results about, as well as further constructions of, proportionally balanced designs are obtained.