The concept of ‘blocking’ in a mathematical sense, seems to have been around for decades. Work in the early 1900's was in the context of topology and set theory and so dealt with infinite sets. See, for instance Bernstein (1908) and Miller (1937). In the 1950's and 1960's, a number of people independently introduced the idea for finite systems. Some of these people were interested in the application to game theory. Other applications have since been introduced in statistics and coding theory. We talk about these applications in section 8.6.

Definition and examples

Let S = (P,L) be a near-linear space. A blocking set in S is a subset B of P such that for each line ℓ ∈ L, 0 < |ℓ ∩ B| < v(ℓ).

Example 8.1.1. Consider figure 1.2.1 of chapter 1. Blocking sets are {3,4}, {1,2,5}. In fact it is easy to see that these are the only blocking sets. Note that if we restrict the space to P′ = {1,2,3,4,5}, the set of all points on at least one line, then the blocking sets above are the complements of each other in P′.

Example 8.1.2. Figure 1.4.1 has {1,3,4,6} as a blocking set. However, this is no longer a blocking set in the linear space of figure 2.1.1. (See exercise 8.7.4.)