This book is essentially a text book that introduces the geometrical objects which arise in the study of vector spaces over finite fields. It advances rapidly through the basic material, enabling the reader to consider the more interesting aspects of the subject without having to labour excessively. There are over a hundred exercises which contain a lot of content not included in the text. This should be taken into consideration and even though one may not wish to try to solve the exercises themselves, they should not be ignored. There are detailed solutions provided to all the exercises.

The first four chapters treat the algebraic and geometric aspects of finite vector spaces. The following three chapters consist of combinatorial applications. There is a chapter containing a brief treatment of applications to groups, real geometry, codes, graphs, designs and permutation polynomials. Then there is a chapter that gives a more in-depth treatment of applications to extremal graph theory, specifically the forbidden subgraph problem, and then a chapter on maximum distance separable codes.

This book is self-contained in the sense that any theorem or lemma which is subsequently used is proven. The only exceptions to this are Bombieri's theorem and the Huxely–Iwaniec theorem concerning the distribution of primes, which are used in the chapter on the forbidden subgraph problem, the Hasse– Weil theorem, which is used to bound the number of points on a plane algebraic curve at the end of the chapter on maximum distance separable codes, and Hilbert's Nullstellensatz, which is used in the appendix on commutative algebra. Although there are almost no prerequisites, it would be helpful to have studied previously some basic algebra and linear algebra, since otherwise the first couple of chapters may appear somewhat brief. There are some theorems that are quoted without proof, but in all cases these appear at the end of some branch and are not built upon.

In this chapter the basic algebraic objects of a group, a ring and a field are defined. It is shown that a finite field has q elements, where q is a prime power, and that there is a unique field with q elements. We define an automorphism of a field and introduce the associated trace and norm functions. Some lemmas related to these functions are proven in the case that the field is finite. Finally, some additional results on fields are proven which will be needed in the subsequent chapters.

Rings and fields

A group G is a set with a binary operation ◦ which is associative ((a ◦ b) ◦ c = a ◦ (b ◦ c)), has an identity element e (a ◦ e = e ◦ a = a) and for which every element of G has an inverse (for all a, there is a b such that a ◦ b = b ◦ a = e). A group is abelian if the binary operation is commutative (a ◦ b = b ◦ a).

A commutative ring R is a set with two binary operations, addition and multiplication, such that it is an abelian group with respect to addition with identity element 0, and multiplication is commutative, associative and distributive (a(b + c) = ab + ac) and has an identity element 1.

The set of integers ℤ is an example of a commutative ring.

An ideal a of a ring R is an additive subgroup with the property that ra ∈ a for all r ∈ R and a ∈ a. For example, the multiples of an element r ∈ R form an ideal, which is denoted by (r).

A coset of a is a set r + a = {r + a | a ∈ a}, for some r ∈ R.

Introduction

In 1852 Francis Guthrie asked whether the regions of every planar map can be coloured with four colours in such a way that no two regions of the map with common boundary receive the same colour. In effect, by duality Guthrie was asking whether every planar graph is 4-colourable. This easily stated problem became known as the famous four-colour problem. Attempts to solve it led to many important results in graph theory, but the problem itself remained unsolved for more than a century. It was finally answered in the positive by Appel and Haken [9], [10], [12]. More about its proof comes later in this chapter.

For more than a century, the four-colour problem was one of the driving forces that led to developments in graph theory, and specifically graph colouring theory. Its generalization – the Heawood problem – was the main motivation for developments in topological graph theory (see [51] or [49]). Both of these were originally formulated as map-colouring problems, asking about colouring the faces of an embedded graph so that adjacent faces receive different colours. Every map-colouring problem can be expressed as a graph-colouring problem by considering the dual graph of the map. Its vertices correspond to the faces of the map, and two vertices are adjacent if the corresponding faces are adjacent on the surface.

In this chapter we discuss the colouring of graphs embedded on surfaces. We start by describing some of the ideas in the proof of the four-colour theorem and give a proof of the list-colouring version of the five-colour theorem that is due to Thomassen.

Introduction

In this introductory section we give the most important definitions required to study hypergraph colouring, and briefly survey the half-century history of this topic. For more details on the material of Sections 1 and 2 we refer to Berge [8], Zykov [76] and Duchet [27].

Let V = {v1, v2, …, vn} be a finite set of elements called vertices, and let ℇ = {E1, E2, …, Em} be a family of subsets of V called edges or hyperedges. The pair ℌ = (V, ℇ) is called a hypergraph with vertex-set V = V(ℌ) and edge-set ℇ = ℇ(ℌ). The hypergraph ℌ = (V, ℇ) is sometimes called a set system. If each edge of a hypergraph contains precisely two vertices, then it is a graph. As in graph theory, the number |V| = n is called the order of the hypergraph. Edges with fewer than two elements are usually allowed, but will be disregarded here. Thus, throughout this chapter we assume that each edge E ∈ ℇ contains at least two vertices, unless stated explicitly otherwise. Edges that coincide are called multiple edges.

In a hypergraph, two vertices are said to be adjacent if there is an edge containing both of these vertices. The adjacent vertices are sometimes called neighbours of each other, and the set of neighbours of a given vertex v is called the (open) neighbourhood N(v) of v. If v ∈ E, then the vertex v and the edge E are incident with each other. For an edge E, the number |E| is called the size or cardinality of E.