Combinatorics of Finite Geometries is an introductory text on the combinatorial theory of finite geometry. Assuming only a basic knowledge of set theory and analysis, it provides a thorough review of the topic and leads the student to results at the frontiers of research. This book begins with an elementary combinatorial approach to finite geometries based on finite sets of points and lines, and moves into the classical work on affine and projective planes. Later, it addresses polar spaces, partial geometries, and generalized quadrangles. The revised edition contains an entirely new chapter on blocking sets in linear spaces, which highlights some of the most important applications of blocking sets--from the initial game-theoretic setting to their very recent use in cryptography. Extensive exercises at the end of each chapter insure the usefulness of this book for senior undergraduate and beginning graduate students.
Author(s): Lynn Margaret Batten
Edition: 2
Publisher: Cambridge University Press
Year: 1997
Language: English
Pages: 208
Contents......Page 8
Preface......Page 12
Preface to the first edition......Page 14
1.1 Some basic concepts: consistency and dependence......Page 16
1.2 Near-linear spaces......Page 19
1.3 New near-linear spaces from old......Page 21
1.4 Dimension......Page 24
1.5 Incidence matrices......Page 26
1.6 The connection number......Page 29
1.7 Linear functions......Page 32
1.8 Exercises......Page 34
2.1 Examples......Page 38
2.2 The de Bruijn-Erdos theorem......Page 40
2.3 Numerical properties......Page 42
2.4 The exchange property......Page 45
2.5 Hyperplanes......Page 47
2.6 Linear functions......Page 49
2.7 Exercises......Page 52
3.1 Projective planes......Page 56
3.2 Finite projective planes......Page 58
3.3 Embedding a near-linear space in a projective plane......Page 59
3.4 Subplanes......Page 60
3.5 Collineations in projective planes......Page 61
3.6 The Desargues configuration......Page 64
3.7 Construction of projective planes from vector spaces......Page 66
3.8 The Pappus configuration......Page 72
3.9 Projective spaces......Page 75
3.10 Desargues configurations again......Page 78
3.11 Exercises......Page 79
4.1 Affine planes......Page 82
4.2 Finite affine planes......Page 84
4.3 Embedding an affine plane in a projective plane......Page 85
4.4 Collineations in affine planes......Page 86
4.5 The Desargues configuration in affine planes......Page 92
4.6 Co-ordinatization in affine planes and the Pappus configuration......Page 94
4.7 Affine spaces......Page 97
4.8 Exercises......Page 100
5.1 The definition......Page 104
5.2 Absolute points......Page 106
5.3 Quadrics......Page 109
5.4 Linear subspaces......Page 114
5.5 Irreducibility......Page 118
5.6 Projective spaces inside polar spaces......Page 120
5.7 A history of polar spaces......Page 122
5.8 Exercises......Page 124
6.1 Definition and some basic results......Page 127
6.2 All known examples......Page 133
6.3 Some combinatorial properties......Page 135
6.4 Generalized quadrangles with s = I = 3......Page 139
6.5 Subquadrangles......Page 142
6.6 Collineations of generalized quadrangles......Page 146
6.7 A brief history of generalized quadrangles......Page 149
6.8 Exercises......Page 150
7.1 The definition......Page 153
7.2 A method of constructing proper partial geometries......Page 156
7.3 Strongly regular graphs......Page 157
7.4 Subgeometries......Page 162
7.5 Pasch's axiom......Page 164
7.6 A history of partial geometries......Page 169
7.7 Exercises......Page 170
8.1 Definition and examples......Page 173
8.2 Blocking sets in projective planes......Page 174
8.3 Blocking sets in affine planes......Page 178
8.4 Blocking sets in Steiner systems......Page 180
8.5 Blocking sets in generalized quadrangles and partial geometries......Page 182
8.6 Applications of blocking sets......Page 186
8.7 Exercises......Page 188
Bibliography......Page 191
Index of notation......Page 205
Subject index......Page 206
