Graph Theory and Additive Combinatorics

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.

1. A bridge between graph theory and additive combinatorics.
2. Forbidding a subgraph I: Mantel's theorem and Turán's theorem.
3. Forbidding a subgraph II: complete bipartite subgraph.
4. Forbidding a subgraph III: algebraic constructions.
5. Forbidding a subgraph IV: dependent random choice.
6. Szemerédi's graph regularity lemma I: statement and proof.
7. Szemerédi's graph regularity lemma II: triangle removal lemma.
8. Szemerédi's graph regularity lemma III: further applications.
9. Szemerédi's graph regularity lemma IV: induced removal lemma.
10. Szemerédi's graph regularity lemma V: hypergraph removal and spectral proof.
11. Pseudorandom graphs I: quasirandomness.
12. Pseudorandom graphs II: second eigenvalue.
13. Sparse regularity and the Gree-Tao theorem.
14. Graph limits I: introduction.
15. Graph limits II: regularity and counting.
16. Graph limits III: compactness and applications.
17. Graph limits IV: inequalities between subgraph densities.
18. Roth's theorem I: Fourier analytic proof over finite field.
19. Roth's theorem II: Fourier analytic proof in the integers.
20. Roth's theorem III: polynomial method and arithmetic regularity.
21. Structure of set addition I: introduction to Freiman's theorem.
22. Structure of set addition II: groups of bounded exponent and modeling lemma.
23. Structure of set addition III: Bogolyubov's lemma and the geometry of numbers.
24. Structure of set addition IV: proof of Freiman's theorem.
25. Structure of set addition V: additive energy and Balog-Szemerédi-Gowers theorem.
26. Sum-product problem and incidence geometry.