## Topics covered

• Lecture 1: Upper bound for R(t,t). Upper bound for R(s,t). Lower bound for R(t,t). Applications of Hypergraph Ramsey to finding convex sets using 3-graphs and 4-graphs.
• Lecture 2: Hypergraph Ramsey with k-colors reduces to 2 colors. 2-color Hypergraph Ramsey. Upper and lower bound for Ramsey numbers of triangles and k colors. Schur's theorem.
• Lecture 3: Goodman's theorem. Graph Ramsey numbers of Clique vs Tree. Ramsey number of clique/induced-star/induced-path in connected graphs. Induced Ramsey Theorem (without proof).
• Lecture 4: Two proofs of the Erdos Szekeres lemma. Some exact Ramsey numbers. van-der Waerdan's Theorem for 2-colors/3-term AP.
• Lecture 5: Proof of van-der Waerdan's Theorem.
• Lecture 6: Ramsey version of Szemeredi's Affine Cube Lemma. Density version of Affine Cube Lemma. Szemeredi's theorem (without proof).
• Lecture 7: Behrend's construction. Lower bound for density version of replete Affine d-Cubes. Roth's Theorem using Triangle Removal Lemma.
• Lecture 8: Density version of homogenous equations with coefficients summing to 0. Deducing Szemeredi's Theorem from the Hypergraph Removal Lemma (without proof). Solution of home assignment 1.
• Lecture 9: High girth and high chromatic number. Erdos's local-vs-global coloring Theorem.
• Lecture 10: Local-vs-global 2-colorability in dense graphs.
• Lecture 11: Local-vs-global 3-colorability in dense graphs. Solution of home assignments.
• Lecture 12: Choice number of planar graphs (Thomassen's Theorem). Orientations, kernels, and choosability. Choice number of planar bipartite graphs.
• Lecture 13: Choice number of degenerate bipartite graphs. Examples of planar graphs with high choice number. The chromatic index of bipartite graphs (Konig's Theorem).
• Lecture 14: Stable Matchings and the Gale-Shapley Theorem. The list chromatic index of bipartite graphs (Dinitz Problem). The list coloring conjecture and Kahn's Theorem (without proof). Long cycles/paths in color-critical graphs.
• Lecture 15: Konig's Infinity Lemma. The de-Bruijn-Erdos Theorem. Solution of home assignments.
• Lecture 16: Perfect graphs. Kovari-Sos-Turan Theorem.
• Lecture 17: Application of the KST Theorem to unit distance problem. Algebraic proof of the Weak Perfect Graph Theorem (Lovasz Theorem).
• Lecture 18: Erdos-Stone Theorem via the hypergraph version of the KST-Theorem. Sperner's Theorem using Hall and via LYM Inequality. The Littlewood-Offord Problem.
• Lecture 19: Odd-town Theorem. Even-town Theorem. 2-distance sets.
• Lecture 20: Nagy's Ramsey construction. Restricted intersections, Frankl-Wilson Theorem. Modular restricted intersections, Deza-Frankl-Singhi Theorem. Frankl-Wilson Ramsey construction. Lindstrom's Theorem.
• Lecture 21: Missing intersection Theorem. Chromatic number of R^n.
• Lecture 22: Kahn-Kalai counter example to Borsuk's conjecture. Bollobas's Theorem.
• Lecture 23: Home assignments. Intersection theorem for uniform sets (Blokhuis trick). Graham-Pollack Theorem.
• Lecture 24: Dual Fischer Inequality. Partitioning K_n into disjoint cliques. Number of lines determined by non-collinear points. Theorem's of Radon, Helly and Caratheodory.
• Lecture 25: Decomposing K_10 into 3 Peterson graphs. Algebraic proof of Bollobas's Theorem. Application to Helly-type property of hypergraph vertex-cover.
• Lecture 26: Hoffman-Singleton Theorem. Bondy's Theorem.
• Lecture 27: Dehn's Theorem (Hilbert's third problem) and the Bolyai-Gerwien Theorem.