18.225 | Fall 2023 | Graduate

Graph Theory and Additive Combinatorics

Lecture Videos

These videos were recorded for 18.217 in 2019. (This course was previously numbered 18.217.)

Lecture 1: A bridge between graph theory and additive combinatorics
Lecture 2: Forbidding a Subgraph I: Mantel’s Theorem and Turán’s Theorem
Lecture 3: Forbidding a Subgraph II: Complete Bipartite Subgraph
Lecture 4: Forbidding a Subgraph III: Algebraic Constructions
Lecture 5: Forbidding a Subgraph IV: Dependent Random Choice
Lecture 6: Szemerédi’s Graph Regularity Lemma I: Statement and Proof
Lecture 7: Szemerédi’s Graph Regularity Lemma II: Triangle Removal Lemma
Lecture 8: Szemerédi’s Graph Regularity Lemma III: Further Applications
Lecture 9: Szemerédi’s Graph Regularity Lemma IV: Induced Removal Lemma
Lecture 10: Szemerédi’s Graph Regularity Lemma V: Hypergraph Removal and Spectral Proof
Lecture 11: Pseudorandom Graphs I: Quasirandomness
Lecture 12: Pseudorandom Graphs II: Second Eigenvalue
Lecture 13: Sparse Regularity and the Green-Tao Theorem
Lecture 14: Graph Limits I: Introduction
Lecture 15: Graph Limits II: Regularity and Counting
Lecture 16: Graph Limits III: Compactness and Applications
Lecture 17: Graph Limits IV: Inequalities between Subgraph Densities
Lecture 18: Roth’s Theorem I: Fourier Analytic Proof over Finite Field
Lecture 19: Roth’s Theorem II: Fourier Analytic Proof in the Integers
Lecture 20: Roth’s Theorem III: Polynomial Method and Arithmetic Regularity
Lecture 21: Structure of Set Addition I: Introduction to Freiman’s Theorem
Lecture 22: Structure of Set Addition II: Groups of Bounded Exponent and Modeling Lemma
Lecture 23: Structure of Set Addition III: Bogolyubov’s Lemma and the Geometry of Numbers
Lecture 24: Structure of Set Addition IV: Proof of Freiman’s Theorem
Lecture 25: Structure of Set Addition V: Additive Energy and Balog-Szemerédi-Gowers Theorem
Lecture 26: Sum-Product Problem and Incidence Geometry

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Fall 2023
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