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.)

This text is designed as an introduction to finite geometry for the undergraduate student. It could be used for second or third year students with some aptitude for, but not necessarily a great deal of background in, mathematics. A second year general student would have a good foundation in synthetic geometry with the completion of the first four chapters. A third year honours student or a fourth year student could be expected to complete the book in a one year course.

As far as background is concerned, only a fundamental knowledge of functions and set theory to the first year university level is essential. Some linear algebra and field theory would be useful for some parts of chapters 5, 6 and 7, but at a minimal level.

Listed in the last section of each chapter are forty to fifty exercises. Some of these are designed to consolidate the student's acquaintance with the concepts presented in the chapter. Others give additional results and introduce new concepts. The exercises form an important part of the material and I would strongly advise that students be assigned several each week. The very difficult problems have been starred.

The material presented in the book is based on the notion of ‘connection number’ in a near-linear space. Given a system of points and lines, for any point p not on a line ℓ, the connection number, c(p,ℓ), is the number of points on ℓ which are connected to p by a line. In chapter 1 (near-linear spaces) there are no restrictions on c(p,ℓ).

The principal changes to the second edition occur in sections 4.5 (which has been completely re-done) and 6.2 (which has been brought up to date). I want to thank particularly Stan Payne for his help in reviewing the appropriate literature for inclusion in 6.2.

In addition, a new chapter, chapter 8, on blocking sets, has been added. Blocking sets have numerous applications in game theory and in testing of statistical designs. We also describe some very recent applications to cryptography.

I wish to thank the University of Manitoba at this time for its support in the production of this new edition.

In this, the final chapter, we consider a generalization of the concept of generalized quadrangle. Generalized quadrangles were first introduced by Tits (1959). The generalization, partial geometry, first appeared independently of Tits' work in a paper by Bose (1963).

The definition

A partial geometry is a finite near-linear space S = (P, L) such that

PG1 for each point p and line ℓ, p∉ℓ implies c(p, ℓ) = α,

PG2 each line has s + 1 points,

PG3 each point is on t + 1 lines,

where α, s and t are fixed positive integers.

Note that Ø is trivially a partial geometry. If S is a partial geometry

which is not Ø, we say S has parameters α, s and t.

We note also that, as for generalized quadrangles, the dual of a partial geometry is again a partial geometry.

Clearly, if α = 1 and GQ2 is satisfied, then S is a generalized quadrangle.

It is not difficult to show that if S ≠ Ø, and α = 1, then GQ2 is satisfied, and we leave this as an exercise.

From the definition of α, s and t, we see that α < s + 1 and α < t + 1. So α ≤ min {s, t} + 1.

For most of this chapter (sections 3.1–3.8) we shall be dealing with the notion of projective plane, before introducing the more general notion of projective space.

Projective planes

Recall (section 2.5) that a projective plane is a linear space in which

1. PP1 any two lines meet,

2. PP2 there exists a set of four points no three of which are collinear.

It follows from L2 that any two lines meet in a unique point.

So far we have seen two examples: the Fano plane (figure 1.1.1) and the extended real plane (example 2.1.3). We present one other example in this section, but will see many more examples later in the chapter, along with examples of more general projective spaces.

The linear space of figure 3.1.1 is a projective plane with thirteen points {1,2,…,13} and thirteen lines {{1,2,3,11}, {4,5,6,11}, {7,8,9,11}, {1,4,7,13}, {2,5,8,13}, {3,6,9,13}, {1,5,9,12}, {2,6,7,12}, {3,4,8,12}, {1,6,8,10}, {2,4,9,10}, {3,5,7,10}, {10,11,12,13}}.

We also recall two lemmas from section 2.5 (lemmas 2.5.3 and 2.5.5) reproducing them here as lemmas 3.1.1 and 3.1.2.

Lemma 3.1.1. A projective plane has the exchange property.

Lemma 3.1.2. A projective plane has dimension 2.

Lemma 3.1.2 gives us the ‘right’ to call a projective plane a plane.

Recalling the labelling of points and lines from chapter 2, we are able to prove the next lemma.

Some basic concepts: consistency and dependence

The understanding throughout this book is that we work with a set P whose elements are called points, and a set L of certain subsets of P, whose elements are called lines. We remind the reader that by definition a set has distinct elements.

A space S = (P, L) is a system of points P and lines L such that certain conditions or axioms are satisfied. We can then consider two points of view: given a system of axioms about points and lines, can we find any spaces which satisfy it, or, given a familiar space (for example, real 3-space), what system or systems of axioms can be used to define it? We are only interested here in the former of these two questions.

As we shall be working a great deal with axiom systems, we discuss some of their properties in this section.

An axiom system is said to be consistent if it is possible to construct an example of a structure satisfying all the axioms.

Introduction

In this chapter we consider semiaffine linear spaces of a much more general nature than in the previous chapter. Also, our methods are quite different. We take a somewhat graph-theoretic approach, using terminology and some basic notions from that area. For us, the vertices of the graph are the lines of the linear space.

Our principle result (Theorem 5.3.3) can be interpreted as follows. If the order n of the linear space S is large enough with respect to the difference between the maximum point degree and the minimum line degree in S, then S embeds in a projective plane of order n. This chapter is based on Beutelspacher and Metsch (1986, 1987).

Our first result is a very general one. The proof is broken into several stages by using propositions. For lines L and H, we denote by m(L,H) the number of lines missing both L and H.

Theorem 5.1.1. Let S = (p,ℒ) be a linear space. Suppose there are a line H and integers a, c, d, e, n and x with the following properties.

1. (1) The degree of H is n + 1 − d > 0.

2. (2) The number of lines missing H is nd + x > 0.

3. (3) For every line L missing H we have n − 1 + a ≤ m(L, H) ≤ n − 1 + c.

4. (4) For any two intersecting lines L1 L2 missing H, we have m(L1,L2) ≤ e + 1.

5. […]

The definition

Given an arbitrary linear space S = (p, ℒ), there is a natural way of introducing further structure on S by distinguishing subsets of p with the property that any element of ℒ containing at least two points of such a subset must be wholly contained within the subset. We call any subset of p with the above property, along with the induced lines, a subspace of 5. Clearly φ and S themselves are subspaces. Moreover, the intersection of any set of subspaces is again a subspace (the empty intersection being S), and so a closure space is induced on S, the closure of any set X of points of p being the intersection of all subspaces containing X.

There are various ways in which a ‘dimension’ can now be assigned to subspaces. In any case, we normally wish to assign dimensions 0 and 1 to the points and lines respectively, and require that V ⊂ W implies that the dimension of V be less than or equal to the dimension of W. We shall make the following definition.

The following pages contain descriptions of the linear spaces on at most nine points. For each linear space, we give a picture, the number of lines, the collineation group acting on the space (this appears in a box) and the number of point orbits under this group.

We use the following group theoretic notations for the collineation groups:

Cn is the cyclic group of order n,

Sn is the symmetric group on n letters,

D2n is the dihedral group of order 2n,

пiH is the direct product of / groups H,

ΣiH is the direct sum of / groups H,

Gn is a group of order n.

The general notation G is used when the group is not describable as a direct sum or product of the cyclic, symmetric or dihedral groups.

We wish to thank Jean Doyen, who, with much time and effort, compiled this comprehensive table for us and offered to let us use it as an appendix to this monograph. Any errors in it are due to the present authors.