In this chapter we give an introduction to a large and important area of combinatorial theory which is known as design theory. The most general object that is studied in this theory is a so-called incidence structure. This is a triple S = (P, B, I), where:

1. P is a set, the elements of which are called points;

2. B is a set, the elements of which are called blocks;

3. I is an incidence relation between P and B (i.e. I ⊆ P × B). The elements of I are called flags.

If (p, B) ∈ I, then we say that point p and block B are incident. We allow two different blocks B1 and B2 to be incident with the same subset of points of P. In this case one speaks of ‘repeated blocks’. If this does not happen, then the design is called a simple design and we can then consider blocks as subsets of P. In fact, from now on we shall always do that, taking care to realize that different blocks are possibly the same subset of P. This allows us to replace the notation (p, B) ∈ I by p ∈ B, and we shall often say that point p is ‘in block B’ instead of incident with B.

It has become customary to denote the cardinality of P by v and the cardinality of B by b.