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18.217 Graph Theory and Additive Combinatorics

18.217 Graph Theory and Additive Combinatorics (Fall 2019, MIT OCW). Instructor: Professor Yufei Zhao. This course examines classical and modern developments in graph theory and additive combinatorics, with a focus on topics and themes that connect the two subjects. The course also introduces students to current research topics and open problems. (from ocw.mit.edu)

Lecture 05 - Forbidding a Subgraph IV: Dependent Random Choice

Professor Zhao discusses in this lecture the dependent random choice method in probabilistic combinatorics and its application to upper bounding the number of edges in an H-free graph, where H is a sparse bipartite graph. Also discussed in this lecture are the problems of forbidding an even cycle, and forbidding a clique 1-subdivision.


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Lecture 01 - A Bridge between Graph Theory and Additive Combinatorics
Lecture 02 - Forbidding a Subgraph I: Mantel's Theorem and Turan's Theorem
Lecture 03 - Forbidding a Subgraph II: Complete Bipartite Subgraph
Lecture 04 - Forbidding a Subgraph III: Algebraic Constructions
Lecture 05 - Forbidding a Subgraph IV: Dependent Random Choice
Lecture 06 - Szemeredi's Graph Regularity Lemma I: Statement and Proof
Lecture 07 - Szemeredi's Graph Regularity Lemma II: Triangle Removal Lemma
Lecture 08 - Szemeredi's Graph Regularity Lemma III: Further Applications
Lecture 09 - Szemeredi's Graph Regularity Lemma IV: Induced Removal Lemma
Lecture 10 - Szemeredi's Graph Regularity Lemma V: Hypergraph Removal and Spectral Proof
Lecture 11 - Pseudo-random Graph I: Quasirandomness
Lecture 12 - Pseudo-random Graph 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-Szemeredi-Gowers Theorem
Lecture 26 - Sum Product Problem and Incidence Geometry