MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 (English)

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Lecture 1: 1. A bridge between graph theory and additive combinatorics

Lecture 2: 2. Forbidding a subgraph I: Mantel's theorem and Turán's theorem

Lecture 3: 3. Forbidding a subgraph II: complete bipartite subgraph

Lecture 4: 4. Forbidding a subgraph III: algebraic constructions

Lecture 5: 5. Forbidding a subgraph IV: dependent random choice

Lecture 6: 6. Szemerédi's graph regularity lemma I: statement and proof

Lecture 7: 7. Szemerédi's graph regularity lemma II: triangle removal lemma

Lecture 8: 8. Szemerédi's graph regularity lemma III: further applications

Lecture 9: 9. Szemerédi's graph regularity lemma IV: induced removal lemma

Lecture 10: 10. Szemerédi's graph regularity lemma V: hypergraph removal and spectral proof

Lecture 11: 11. Pseudorandom graphs I: quasirandomness

Lecture 12: 12. Pseudorandom graphs II: second eigenvalue

Lecture 13: 13. Sparse regularity and the Green-Tao theorem

Lecture 14: 14. Graph limits I: introduction

Lecture 15: 15. Graph limits II: regularity and counting

Lecture 16: 16. Graph limits III: compactness and applications

Lecture 17: 17. Graph limits IV: inequalities between subgraph densities

Lecture 18: 18. Roth's theorem I: Fourier analytic proof over finite field

Lecture 19: 19. Roth's theorem II: Fourier analytic proof in the integers

Lecture 20: 20. Roth's theorem III: polynomial method and arithmetic regularity

Lecture 21: 21. Structure of set addition I: introduction to Freiman's theorem

Lecture 22: 22. Structure of set addition II: groups of bounded exponent and modeling lemma

Lecture 23: 23. Structure of set addition III: Bogolyubov's lemma and the geometry of numbers

Lecture 24: 24. Structure of set addition IV: proof of Freiman's theorem

Lecture 25: 25. Structure of set addition V: additive energy and Balog-Szemerédi-Gowers theorem

Lecture 26: 26. Sum-product problem and incidence geometry