8 - Blocking sets
Published online by Cambridge University Press: 26 March 2010
Summary
Lyke as a huntsman after weary chace,
Seeing the game from him escapt away,
Sits downe to rest him in some shady place,
With panting hounds beguiled of their prey:
So, after long pursuit and vaine assay,
When I all weary had the chace forsooke,
The gentle deare returned the selfe-same way,…
Edmund Spenser Amoretti Sonnet LXVIIThe 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.)
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- Combinatorics of Finite Geometries , pp. 158 - 175Publisher: Cambridge University PressPrint publication year: 1997