CONTENTS 10.2 Clique Number,183 10.3 Chromatic Number.184 10.4 Zero-One Laws,186 105 Exercises 193 Counting Subgraphs,195 11 The Erdos-Renyi Phase Transition 197 11.1 An Overview.197 11.2 c 3 9 -Watson Branching Process,201 11. Analysis of the Poisson Branching Process.202 11.5 The Graph Branching Model,204 11.6 The Graph and Poisson Processes Compared,205 11.7 The Parametrization Explained.207 118 The Subcritical Regions.208 11.9 The Supercritical Regimes,209 11.10 The Cri ical winde 212 11.12 Exercises,219 Long paths in the supercritical regime,220 12 Circuit Complexity 223 12.1 Preliminaries,223 12.2 Random Restrictions and Bounded-Depth Circuits,225 12.3 More on Bounded-Depth Circuits,229 12.4 Monotone Circuits.232 12.5 Formulae.235 12.6 Exercises,236 Maximal Antichains,237 13 Discrepancy 239 13.1 Basics,239 13. 2 Six Standard Deviations Suffice,241 13. 3 Linear and Hereditary Discrepancy,245 13.4 Lower Bounds.248 13.5 The Beck-Fiala Theorem,250 13.6 Exercises,251 Unbalaneing Lights,253 14 Geometry 255 14.1 The Greatest Angle Among Points in Euclidean Spaces,256 x CONTENTS 10.2 Clique Number, 183 10.3 Chromatic Number, 184 10.4 Zero–One Laws, 186 10.5 Exercises, 193 Counting Subgraphs, 195 11 The Erdos–Rényi Phase Transition 197 ˝ 11.1 An Overview, 197 11.2 Three Processes, 199 11.3 The Galton–Watson Branching Process, 201 11.4 Analysis of the Poisson Branching Process, 202 11.5 The Graph Branching Model, 204 11.6 The Graph and Poisson Processes Compared, 205 11.7 The Parametrization Explained, 207 11.8 The Subcritical Regions, 208 11.9 The Supercritical Regimes, 209 11.10 The Critical Window, 212 11.11 Analogies to Classical Percolation Theory, 214 11.12 Exercises, 219 Long paths in the supercritical regime, 220 12 Circuit Complexity 223 12.1 Preliminaries, 223 12.2 Random Restrictions and Bounded-Depth Circuits, 225 12.3 More on Bounded-Depth Circuits, 229 12.4 Monotone Circuits, 232 12.5 Formulae, 235 12.6 Exercises, 236 Maximal Antichains, 237 13 Discrepancy 239 13.1 Basics, 239 13.2 Six Standard Deviations Suffice, 241 13.3 Linear and Hereditary Discrepancy, 245 13.4 Lower Bounds, 248 13.5 The Beck–Fiala Theorem, 250 13.6 Exercises, 251 Unbalancing Lights, 253 14 Geometry 255 14.1 The Greatest Angle Among Points in Euclidean Spaces, 256