The theory of finite linear spaces

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
Dedication......Page 9
Preface......Page 10
1.1 Definitions and examples......Page 12
1.2 Affine spaces and projective spaces......Page 14
1.3 The connection between affine and projective spaces......Page 21
1.4 The history of finite linear spaces......Page 22
1.5 Some basic results and the Fundamental Theorem......Page 25
1.6 Designs......Page 28
1.7 Another example of a linear space......Page 29
1.8 Exercises......Page 30
1.10 References......Page 31
2.1 Aim of the chapter......Page 33
2.2 The details......Page 34
2.3 The pseudo-complement theorems......Page 36
2.5 Research problems......Page 42
2.6 References......Page 43
3.1 Introduction......Page 44
3.2 Two consecutive line sizes......Page 45
3.3 Three consecutive line sizes......Page 54
3.4 Two non-consecutive line sizes......Page 68
3.5 On the existence of linear spaces with certain line sizes......Page 76
3.6 Exercises......Page 77
3.8 References......Page 78
4.1 1-Semiaffine linear spaces......Page 79
4.2 {0,1}-Semiaffine linear spaces......Page 80
4.3 {0,s,t}-Semiaffine linear spaces with non-constant point degree......Page 83
4.4 {s- l ,s}-Semiaffine linear spaces with constant point degree......Page 91
4.5 {0,1,s}-Semiaffine linear spaces with constant point degree......Page 95
4.6 {0,1,2}-Semiaffine linear spaces......Page 99
4.7 Exercises......Page 105
4.9 References......Page 106
5.1 Introduction......Page 107
5.2 Embedding linear spaces with constant point degree......Page 115
5.3 Embedding linear spaces with arbitrary point degree......Page 117
5.6 References......Page 129
6.1 Restricted linear spaces......Page 130
6.2 The combinatorial approach......Page 136
6.3 The algebraic approach......Page 142
6.4 Exercises......Page 145
6.6 References......Page 146
7.1 The definition......Page 147
7.2 Problems in d-dimensions......Page 148
7.3 7r-Spaces......Page 149
7.4 Local structures......Page 157
7.5 `Global\' conditions......Page 164
7.6 A final embedding theorem......Page 171
7.7 Exercises......Page 175
7.9 References......Page 176
8.1 Reasons for this chapter and some background......Page 178
8.2 Point and line orbits......Page 179
8.3 Relationships between transitivities......Page 183
8.4 Characterization theorems......Page 187
8.7 References......Page 198
Appendix......Page 200
Notation index......Page 223
Subject index......Page 224