Lecture 1: Appetizer: Triangles and Equations §0
Lecture 2: Forbidding a Subgraph: Mantel’s Theorem and Turán’s Theorem §1.1–1.2
Lecture 3: Forbidding a Subgraph: Supersaturation, Kővári-Sós-Turán, Erdős-Stone-Simonovits §1.3–1.5
Lecture 4: Forbidding a Subgraph: Geometric Application, Forbidding Cycles, Dependent Random Choice §1.4, 1.6–1.7
Lecture 5: Forbidding a Subgraph: Randomized and Algebraic Lower Bound Constructions §1.9–1.10
Problem set 1 due
Lecture 6: Graph Regularity: Statement and Proof §2.1
Lecture 7: Graph Regularity: Triangle Counting and Removal Lemmas, Proof of Roth’s Theorem §2.2–2.4
Lecture 8: Graph Regularity: Behrend’s Construction of 3-AP-Free Sets, Graph Counting and Removal, Erdős-Stone-Simonovits Again, Graph Property Testing. §2.5–2.6, 2.9
Lecture 9: Graph Regularity: Induced Removal and Strong Regularity, Hypergraph Removal and Regularity §2.8, 2.10–2.11
Problem set 2 due
Lecture 10: Pseudorandom Graphs: Quasirandom Graphs, §3.1
Lecture 11: Pseudorandom Graphs: Expander Mixing Lemma, Abelian Cayley Graphs and Eigenvalues, Quasirandom Groups, §3.2–3.4
Lecture 12: Pseudorandom Graphs: Quasirandom Cayley Graphs (sketch), Second Eigenvalue Bound §3.5–3.6
Problem set 3 due
Lecture 13: Graph Limits: Introduction, Graphon, Cut Distance §4.1–4.3
Lecture 14: Guest lecture by Fan Chung
Lecture 15: Graph Limits: Homomorphism Density, Counting Lemma, Weak Regularity Lemma, §4.3, 4.5–4.6
Lecture 16: Graph Limits: Martingale Convergence Theorem, Compactness of the Graphon Space, Equivalence of Convergence, §4.7–4.9
Problem set 4 due
Lecture 17: Graph Homomorphism Inequalities: Edge-Triangle Region, Cauchy-Schwarz §5.1–5.2
Lecture 18: Graph Homomorphism Inequalities: Flag Algebra, Hölder, Lagrangian §5.3–5.4
Lecture 19: Forbidding 3-Term Arithmetic Progressions: Finite Field §6.1–6.2
Lecture 20: Forbidding 3-Term Arithmetic Progressions: Integers §6.3–6.4
Problem set 5 due
Lecture 21: Forbidding 3-Term Arithmetic Progressions: Polynomial Method. Structure of Set Addition: Statement of Freiman’s Theorem §6.5, 7.1
Lecture 22: Structure of Set Addition: Ruzsa Triangle Inequality, Plünnecke’s Inequality, Ruzsa Covering Lemma, Freiman’s Theorem in Groups of Bounded Exponent §7.2–7.5
Lecture 23: Structure of Set Addition: Freiman Homomorphisms, Ruzsa’s Model Lemma, Bogolyubov’s Lemma §7.6–7.8
Lecture 24: Structure of Set Addition: Geometry of Numbers, Proof of Freiman’s Theorem, Polynomial Freiman-Ruzsa Conjecture §7.9–7.12
Lecture 25: Structure of Set Addition: Additive Energy and the Balog-Szemerédi-Gowers Theorem §7.13
Problem set 6 due
Lecture 26: Sum-Product Problem §8
Lecture 27: Progressions in Sparse Pseudorandom Sets: the Green-Tao Theorem §9
