Resolvable configurations
In the paper under review, the author introduces a new class of configurations called resolvable. An incidence structure \(\mathcal{S}\) is an ordered triple \(\mathcal{S} = (\mathcal{P},\mathcal{B},I)\) of mutually disjoint sets such that \(\mathcal{P}\) is the set of points, \(\mathcal{B}\) is the set of blocks, and \(I \subseteq \mathcal{P} \times \mathcal{B}\) is an incidence relation. Let \(v\), \(b\), \(r\), \(k\), \(\lambda\) be positive integers, and consider the following conditions: (C1) \(|\mathcal{P}| = v\); (C2) \(|\mathcal{B}| = b\); (C3) each point is incident with precisely \(r\) blocks; (C4) each block is incident with precisely \(k\) points; (C5) two distinct points are incident with at most one block; (C6) every pair of distinct points is incident with precisely \(\lambda\) block. A combinatorial configuration \((v_{r},b_{k})\) is an incidence structure satisfying conditions (C1)--(C4). If the condition (C5) is also satisfied, then the configuration is called lineal. For integers \(v > k \geq 2\), an incidence structure is called a \((v,k,\lambda )\)-balanced incomplete block design if it satisfies conditions (C1)--(C4) and (C6). This name is usually abbreviated to \((v,k,\lambda)\)-BIBD. A geometric configuration is, in the simplest case, an incidence structure in which each \(\mathcal{P}\) and \(\mathcal{B}\) consist of points and (straight) lines, respectively, in the Euclidean or real projective plane, such that the conditions (C1)--(C4) are satisfied. Every geometric configuration of type \((v_{r},b_{k})\) has an underlying combinatorial configuration of the same type. On the other hand, not every combinatorial configuration can be realized as a geometric configuration. Now suppose that \(\mathcal{R} = (\mathcal{P}, \mathcal{B})\) is a \((v,k,\lambda)\)-BIBD with set of points \(\mathcal{P}\) and set of blocks \(\mathcal{B}\). A parallel class in \(\mathcal{R}\) is a set of blocks such that each element of \(\mathcal{P}\) is incident with precisely one block. A partition of \(\mathcal{B}\) into \(r\) parallel classes is called a resolution, and \(\mathcal{R}\) is said to be a resolvable BIBD if \(\mathcal{B}\) has at least one resolution. Now we are going to provide a formal definition of resolvable configurations. Definition. For integers \(r,k \geq 2\), \(v\geq 4\), suppose that \(\mathcal{C} = (\mathcal{P},\mathcal{B})\) is a configuration of type \((v_{r},b_{k})\) with the set of points \(\mathcal{P}\) and the set of blocks \(\mathcal{B}\). A parallel class in \(\mathcal{C}\) is a set of blocks such that each element of \(\mathcal{P}\) is incident with precisely one block. A partition of \(\mathcal{B}\) into \(r'\) parallel classes is called a resolution, and \(\mathcal{C}\) is said to be a resolvable configuration if \(\mathcal{B}\) has at least one resolution. One can show, for instance, that in a resolvable configuration of type \((v_{r},b_{k})\) the number of parallel classes is precisely \(r\). Moreover, the author delivers other meaningful properties of resolvable configurations, and also interesting examples, especially the last example, which is an infinite family of resolvable point-circle configurations, has a very nice geometric background.
scientific article; zbMATH DE number 7349670
scientific article