Independence Theory in Combinatorics

1 Preliminaries. - 1. 1 General introductory and historical remarks. - 1. 2 Sets families and graphs. - 1. 3 Vector spaces; linear and affine independence. - Exercises. - 2 Independence spaces. - 2. 1 Axioms and some basic theorems. - 2. 2 Some induced structures. - 2. 3 Submodular functions. - 2. 4 Sums of independence structures. - Exercises. - 3 Graphic spaces. - 3. 1 The cycle and cutset structures of a graph. - 3. 2 Connections with vector spaces. - 3. 3 Applications of independence theory to graphs. - Exercises. - 4 Transversal spaces. - 4. 1 Hall's theorem and its generalization. - 4. 2 The partial transversals of a family of sets. - 4. 3 Duals of transversal structures. - 4. 4 Extensions of Hall's theorem. - 4. 5 Applications. - Exercises. - 5 Appendix on representability. - 5. 1 Representability in general. - 5. 2 Linear representability. - 5. 3 Induced structures. - 5. 4 Linear representability over specified fields. - 5. 5 Some spaces which are not linearly representable. - Exercises. - Hints and solutions to the exercises. - Further reading. Language: English