ContentsXV7. Extremal Graph Theory1637.1 Subgraphs*1647.2 Minors(*).1697.3 Hadwiger's conjecturei.1727.4 Szemeredi's regularity lemm1751837.5 Applying the regularity le189ExercisesNotes1928.Infinite Graphs.1958.1 Basic notions, facts and techniques1968.2 Paths, trees, and ends(*)2042128.3Homogeneus and universal grap2168.4Connectivityndmatching8.5 The topological end space226Exercises237Notes2449. Ramsey Theory for Graphs2512529.1 Ramsey's original theorems9.2 Ramsey numbers(*)2559.3InducedRamsey theorems2589.4 Ramsey properties and connectivity(*).268Exercises271Notes27210. Hamilton Cycles27510.1Simplesufficientconditions27510.2 Hamilton cycles and degree sequences27828110.3Hamilton cycles in the square of a graphExercises289Notes290Contents xv 7. Extremal Graph Theory ....................................... 163 7.1 Subgraphs* .................................................... 164 7.2 Minors(∗) ....................................................... 169 7.3 Hadwiger’s conjecture* ......................................... 172 7.4 Szemer´edi’s regularity lemma ................................... 175 7.5 Applying the regularity lemma ................................. 183 Exercises ....................................................... 189 Notes .......................................................... 192 8. Infinite Graphs ................................................. 195 8.1 Basic notions, facts and techniques* ............................ 196 8.2 Paths, trees, and ends(∗) ........................................ 204 8.3 Homogeneous and universal graphs* ............................ 212 8.4 Connectivity and matching ..................................... 216 8.5 The topological end space ...................................... 226 Exercises ....................................................... 237 Notes .......................................................... 244 9. Ramsey Theory for Graphs ................................... 251 9.1 Ramsey’s original theorems* .................................... 252 9.2 Ramsey numbers(∗) ............................................. 255 9.3 Induced Ramsey theorems ...................................... 258 9.4 Ramsey properties and connectivity(∗) ........................... 268 Exercises ....................................................... 271 Notes .......................................................... 272 10. Hamilton Cycles .............................................. 275 10.1 Simple sufficient conditions* .................................... 275 10.2 Hamilton cycles and degree sequences* ......................... 278 10.3 Hamilton cycles in the square of a graph ........................ 281 Exercises ....................................................... 289 Notes .......................................................... 290