Symmetric designs, sets with two intersection numbers and Krein parameters of incidence graphs (Q2747193)

A symmetric block design is a triple \((\mathcal{P},\mathcal{B},\mathcal{I})\) where \(\mathcal{P}\) denotes a set of \(v\) points, \(\mathcal{B}\) denotes a set of \(v\) blocks, and \(\mathcal{I} \subseteq \mathcal{P} \times \mathcal{B}\) is an incidence relation with the following properties: any point (resp. block) is incident with exactly \(\lambda\) common blocks (resp. \(\lambda\) common points). The incidence graph of a symmetric design is distance-regular, and thus its adjacency algebra is a Bose-Mesner algebra. The main observation of this paper is the fact that a certain Krein parameter of this association scheme is equal to zero has non-trivial combinatorial consequences for the intersection properties of certain subconfigurations in the design. The author explains how the configurations under study arise from a special class of sets with two intersection numbers, and gives examples of how the intersection theorems apply in the case of finite projective planes and projective geometries. The author hopes that this technique will be applied more widely, for instance in the study of block designs or error-correcting codes.