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Finite Geometry and Combinatorial Applications

Simeon Ball
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Student Texts 82
[Reviewed by
Mark Hunacek
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Back in March of this year, I reviewed for this column the Dover republication of An Introduction to Finite Projective Planes by A. A. Albert and Reuben Sandler (hereafter denoted IFPP). I mentioned in that review that the study of finite projective planes “offers a beautiful combination of geometry, algebra, and combinatorics”. The book now under review, the latest entry in Cambridge University Press’s respected London Mathematical Society Student Texts series, provides further evidence in support of that statement.

In my review of IFPP, I discussed how projective planes can be defined axiomatically, and how a broad class of examples of projective planes can be defined by a field (or more generally a division ring). I also mentioned the connection between these special projective planes and some classical theorems of geometry (those of Desargues and Pappus), and discussed the fact that other projective planes are related to more complicated algebraic structures (ternary rings). Ball’s text is a more modern look at some of these issues (and others); the textual coverage is in some sense narrower than in IFPP, but, in most other respects, considerably broader.

The book under review is narrower than IFPP in the sense that the projective geometries discussed here are all defined by fields, or, more precisely, by vector spaces over fields. (Projective n-space is just an (n+1)-dimensional vector space over a field F; the points in this space are the one-dimensional subspaces, and the lines are the two-dimensional ones. If we specialize to the plane (n = 2) and pick an ordered basis for the vector space, we get the definition given in the review of IFPP.) Ball’s text is also narrower in scope than IFPP because there isn’t a lot of traditional geometry here; Desargues’s theorem is discussed, but many other classical geometric theorems (Pappus, Pascal, etc.) are not: the main focus in the book under review is on combinatorics.

On the other hand, Ball’s text contains a lot of material that is not found in IFPP: projective spaces are discussed (not just, as in IFPP, planes), and in addition to projective spaces the author also discusses polar spaces: these are also defined from vector spaces, but here the vector space is equipped with a quadratic form or a sesquilinear form, and only certain vector subspaces are considered geometric subspaces. Polar spaces are not discussed in the textbook literature nearly as often as are projective spaces; what discussions do exist tend to be in books such as Points and Lines: Characterizing the Classical Geometries by Ernest E. Shult, described in this review as “more suitable as a handbook for experts and would-be experts than as a first introduction to incidence geometry.” Prior to my seeing this text, the only textbook-level source that I was aware of that discussed them was a set of lecture notes by Peter Cameron; in fact, Ball credits these notes as being the inspiration for his own set of notes, on which the book under review is “loosely based”.

The material described above, when suitably fleshed out, corresponds to the first four chapters of this book. Chapters 1 and 2 discuss, respectively, fields (mostly finite) and vector spaces. The material in these chapters is largely a review of the basics, but even though there is little or no new material here, the chapters are still fun to read: the author’s writing style is brisk, efficient, and clear. In chapter 3, which would not be out of place in a textbook on advanced linear algebra, the author discusses forms: bilinear, sesquilinear, quadratic, etc. (Sesquilinear forms are, roughly speaking, forms that are bilinear “up to an automorphism” of the underlying field. A more precise definition, of course, is given in the text, as is a theorem classifying them.)

Geometry is then introduced in chapter 4. First projective spaces are defined, then polar spaces are. Various topics related to these concepts are also discussed.

The writing style in chapters 3 and 4 continues to be brisk and efficient; it is generally clear, though on occasion the author omits qualifying remarks from a statement of a theorem, under the assumption that the reader will know from previous language in the text what is going on. For example, theorem 4.3 reads in its entirety: “For each positive integer \(r\geq 2\), there are six polar spaces over \(\mathbb{F}_q\), up to isomorphism.” On its face, this is confusing, since the integer \(r\) plays no role in the statement of the theorem; however, in the paragraph before the statement of the theorem, it is stated that \(r\) will denote the rank of a polar space. So what the author really means is that there are six polar spaces of rank \(r\) over \(\mathbb{F}_q\). This is admittedly the sort of thing that is not likely to bother a professional, but it might potentially be confusing to a student. (I should also point out that the word “rank” does not appear in the Index to this book, and is not the only term whose absence I noticed.)

The remaining three chapters of the book discuss applications to combinatorics. Specifically, chapter 5 provides a nice survey of finite geometries to various areas in discrete mathematics: block designs, graphs, codes, etc. Then chapters 6 and 7 focus more narrowly on two significant problems in combinatorics: chapter 6 talks about the forbidden subgraph problem and chapter 7 addresses maximum distance separable (MDS) codes.

The material in these last two chapters is fairly technical, but the basic ideas can be at least be (very) broadly described. The forbidden subgraph problem asks: given a graph H and a positive integer n, what is the maximum number of edges (denote this ex(n, H)) that an n-vertex graph can have and not contain a copy of H as a subgraph? Using material from the chapter on polar geometries, the author derives some theorems about the long-term behavior of ex(n, H). As for MDS codes: although the definition of these objects is too involved for this review, suffice it to say that theorems relating to their existence, and information about parameters associated with them, can be proved via certain structures in projective geometries.

All chapters contain exercises, some of which are used to develop interesting topics that were omitted from the body of the text. Affine planes, for example, are defined, and their basic properties studied, in several exercises, some of which discuss their relationship with mutually orthogonal Latin squares. Solutions to all the exercises appear in an appendix.

There are also two other appendices. One contains proofs of some results of the text that were deferred to an appendix to avoid disrupting the flow of the exposition, and the other contains notes, references and attributions. In addition, there is also a rather detailed ten-page bibliography.

A word about the likely audience of this text: a back-cover blurb announces that this book “is ideal for anyone, from a third-year undergraduate to a researcher…” As is often the case with these assertions, this must be taken with a grain of salt, keeping in mind that it is British undergraduates to which this statement applies. On this side of the pond, I doubt very much that average undergraduates will find this text hospitable. I think that the most likely student audience here would consist of graduate students (second year or above), looking to learn some fairly sophisticated combinatorics. This audience, however, should, particularly since books on polar spaces are not exactly thick on the ground, derive considerable benefit from this text, as should professional mathematicians with current or potential research interests in this area.

Mark Hunacek ( teaches mathematics at Iowa State University. 

1. Fields
2. Vector spaces
3. Forms
4. Geometries
5. Combinatorial applications
6. The forbidden subgraph problem
7. MDS codes
Appendix A. Solutions to the exercises
Appendix B. Additional proofs
Appendix C. Notes and references