These notes present an investigation of a condition similar to Euclid's parallel axiom for subsets of finite sets. The background material to the theory of parallelisms is introduced and the author then describes the links this theory has with other topics from the whole range of combinatorial theory and permutation groups. These include network flows, perfect codes, Latin squares, block designs and multiply-transitive permutation groups, and long and detailed appendices are provided to serve as introductions to these various subjects. Many of the results are published for the first time.
Author(s): Peter J. Cameron
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1976
Language: English
Pages: 149
Cover......Page 1
Title......Page 2
Copyright......Page 3
Contents......Page 4
Introduction......Page 6
1. The existence theorem......Page 9
Appendix: The integrity theorem for network flows......Page 17
2. The parallelogram property......Page 24
Appendices: The binary perfect code theorem......Page 31
Association schemes and metrically regular graphs......Page 41
3. Steiner points and Veblen points......Page 51
Appendix: Steiner systems......Page 58
4. Minimal edge-colourings of complete graphs......Page 68
Appendix: Latin squares, SDRs, and permanents......Page 79
5. Biplanes and metric regularity......Page 86
Appendix: Symmetric designs......Page 102
6. Automorphism groups......Page 113
Appendix: Multiply transitive groups......Page 124
7. Resolutions and partition systems......Page 134
Bibliography......Page 143
Index......Page 148
