xviiContents7. Extremal Graph Theor1737.1 Subgraphs*1747.2 Minors(*)807.3 Hadwiger's conjectu1837.4 Szemeredi's regularity le87.5Applyingtheregularitylen195...RyereNotes2048. Infinite Graphs2098.1Bas2108.2 Paths,tres, and ends(*)..2198.3 Homogereneous and universal graphs*2288.4 Connectivity and matching2318.5 R2428.6 Graphs with ends: the complete picture..2458.7 The topological cycle spac25488Infinite graphs aslimitsoffinite258..261ExercisesNotes2739. RamTheory for Grapl2839.1 Ramsey's origina284heoret9.2Ramseynumbers(*)2872909.3InducedRamseythere9.4 Rar300ivit.303Exercises304Note10. Hamilton Cycles30730710.1Sufficientconditions*10.2Hamiltoncvcles311andde10.3Hamilton cycles in the square of a grap314.319Exercises++++++++++.320Notes+.++++Contents xvii 7. Extremal Graph Theory ....................................... 173 7.1 Subgraphs* .................................................... 174 7.2 Minors(∗) ....................................................... 180 7.3 Hadwiger’s conjecture* ......................................... 183 7.4 Szemer´edi’s regularity lemma ................................... 187 7.5 Applying the regularity lemma ................................. 195 Exercises ....................................................... 201 Notes .......................................................... 204 8. Infinite Graphs ................................................. 209 8.1 Basic notions, facts and techniques* ............................ 210 8.2 Paths, trees, and ends(∗) ........................................ 219 8.3 Homogeneous and universal graphs* ............................ 228 8.4 Connectivity and matching ..................................... 231 8.5 Recursive structures ............................................ 242 8.6 Graphs with ends: the complete picture ......................... 245 8.7 The topological cycle space ..................................... 254 8.8 Infinite graphs as limits of finite ones ........................... 258 Exercises ....................................................... 261 Notes .......................................................... 273 9. Ramsey Theory for Graphs ................................... 283 9.1 Ramsey’s original theorems* .................................... 284 9.2 Ramsey numbers(∗) ............................................. 287 9.3 Induced Ramsey theorems ...................................... 290 9.4 Ramsey properties and connectivity(∗) ........................... 300 Exercises ....................................................... 303 Notes .......................................................... 304 10. Hamilton Cycles .............................................. 307 10.1 Sufficient conditions* ........................................... 307 10.2 Hamilton cycles and degree sequences ........................... 311 10.3 Hamilton cycles in the square of a graph ........................ 314 Exercises ....................................................... 319 Notes .......................................................... 320