Combinatorics of Finite Geometries

85 admirably clear. While the book is devoted to a single problem, the techniques employed would seem to have wider utility, and so should appeal to those interested in efficiency of algorithms in graph theory, and further (perhaps) to those wishing to apply such ideas to the real world. I enjoyed reading this book. It is written in a way which should appeal to advanced undergraduate and postgraduate students who wish to learn something of the synthetic approach to finite geometry as well as to practising mathematicians with interests in other branches of the subject. The subject is developed in a most easily followed style beginning with Near Linear Spaces and Linear Spaces; both are then given an axiomatic treatment in the first two chapters. The concept of projective plane is developed in Chapter 3. This includes a discussion of the interrelationship between Desargues' theorem and the study of collineations. After the introduction of coordinates there is work on Pappus' theorem and then the chapter includes a section on projective spaces of dimension greater than two. It concludes with proving that Desargues' theorem holds in any projective plane embedded in a projective w-space, with n ^ 3. In Chapter 4 Affine spaces are considered. Firstly there is a discussion of affine planes, their relationship with projective planes and some particular results concerning collineations in them. The idea of affine spaces is then introduced. Once these various spaces have been introduced, along with some knowledge of their structure, it is possible to discuss geometrical objects contained within them. For a detailed description of the algebraic geometry of Galois spaces one should refer to the comprehensive text by Hirschfeld [2]. In Chapter 5 of Professor Batten's book the ideas of polar spaces are developed axiomatically. The connection with the polarity with respect to a quadric follows, together with linear subspaces of a polar system. The discussion includes a note of the usual results due to Primrose [3] concerning the numbers of subspaces on elliptic and hyperbolic quadrics in Galois spaces. This chapter concludes with a short history of the ideas of polar spaces, stating various results due to Tits [4] and Buekenhout and Shult [1]. The next chapter is devoted to generalized quadrangles. Some basic results concerning the relationships between the parameters are first proved, then the seven known classes of these structures are described. A number of further results concerning these …