An affine plane of order n has n2 + n lines, any two of which are either parallel or intersecting. The relation of parallelism on the set of lines is an equivalence relation, and so it partitions the set of lines into n + 1 parallel classes of cardinality n. Each point lies on exactly one line from each parallel class. The block set of the complement of an affine plane of order n can be partitioned into n + 1 classes so that each point is contained in exactly n – 1 blocks from each class. Similar partitions exist in affine geometries of higher dimension. In this chapter we study a more general notion of resolution of an incidence structure, i.e., a partition of the block set of the structure into classes so that each point is contained in a constant number of blocks from each class.

Bose's Inequality

The incidence structures on which we define the notion of resolution are pairwise balanced designs.

Definition 5.1.1. Let λ be a positive integer. A pairwise balanced design (PBD) of index λ is an incidence structure D = (X, B, I) such that X ≠ Ø, every x ∈ X is incident with more than λ blocks, and, for any distinct x, y ∈ X, there are precisely λ blocks that are incident with both x and y. If a PBD of index λ has constant replication number r, it is called an (r, λ)-design.