Math 583: This is a redesigned course: It includes covering in depth the Szemeredi Regularity Lemma and the Hypergraph Container Method. There will be not much on posets or matroids, the official syllabus of the course will be updated at one point.
The lecture notes cover one semester of classes, consisting of 43 classes, each 50 minutes (the number of Lecture notes is 33, as many of them requires more than one class to be covered).
Announcements:
Lecture I: Motivation, Definitions and the statement of the Szemerédi Regularity Lemma. Preparation for the proof.
Lecture II: Szemerédi Regularity Lemma: Definitions, proof. Slicing Lemma. Embedding Lemma.
Lecture III: Proof of Erdõs-Stone Theorem using Turán Theorem and Szemerédi Regularity Lemma. Counting K_r-free graphs, stability version. Ramsey function R(G,G) is linear for bounded degree graphs.
Lecture IV: Counting Lemma, Removal lemma, (6,3)-Theorem, Induced Matching, Equivalence of (6,3), Induced Matching, 3-AP Theorems.
Lecture V: Corner Theorem, Corner implies Szemerédi Theorem, weak hypergraph regularity lemma, simplex removal lemma, Behrend construction.
Lecture VI: Multicolored Regularity Lemma, Degree version, Blow-up lemma, Sparse regularity lemma, Alon-Fischer-Krievelich-Szegedy Regularity lemma, Number of induced C_4-free graphs.
Lecture VII: Property testing (triangle-free), Ramsey-Turan K_4-free, number of edges of critical 2-diameter graphs. Caccetta-Haggvist Conjecture.
Lecture VIII: Number of triangles in K_{1,2,2}-free graphs, Axenovich: edge-colorings with at least 9 different colors in every K_5, the number of maximal triangle-free graphs.
Lecture IX: Volume computing of metric polytope. Luczak: minimum degree > n/3, triangle-free implies bounded chromatic number.
Lecture X: Sudakov-Tomon: the extremal number of bipartite K_{t,t}-free graph H, with one side having maximum degree t.
Lecture XI: Container introduction, graph container algorithm, number of q-colorings of d-regular graphs. typed notes
Lecture XII: Alon-Friedgut-Kalai-Kindler: Independence number of random subgraphs of a graph. Number of antichains in Boolean lattice. typed notes.
Lecture XIII: Sperner Theorem, supersaturation. Balogh-Mycroft-Treglown: random variant of Sperner Theorem. typed notes
Lecture XIV: The number of Sidon sets. typed notes.
Lecture XV: The number of C_4-free graphs (Kleitman-Winston). typed notes.
Lecture XVI: List coloring (from Alon-Spencer book). typed notes.
Lecture XVII: Applications of container lemma: Random Szemerédi, Random Turán Theorems. typed notes.
Lecture XVIII: Number of Maximal Triangle-free graphs. Volume of Metric Polytopes. typed notes.
Lecture XIX: (3,4)-problem, epsilon-nets. typed notes.
Lecture XX: Random Ramsey numbers, Folkman numbers. typed notes.
Lecture XXI: 3-uniform hypergraph container lemma, via Balogh-Wagner. typed notes.
Lecture XXII: k-uniform hypergraph container lemma. typed notes.
Lecture XXIII: Saxton - Thomason: Simple probabilistic proof for linear hypergraphs. typed notes.
Lecture XXIV: Number of Maximal sumfree sets (based on BLST). typed notes.
Lecture XXV: Proof of the KLR-Conjecture (based on BMS). typed notes.
Lecture XXVI: Ferber-McKinley-Samotij: supersaturation; to count bipartite graphs. typed notes.
Lecture XXVII: Number of independents sets in the hypercube. typed notes.
Lecture XXVIII: Edge-random EKR-Theorem typed notes.
Lecture XXIX: Introduction to Fourier method, number of monochromatic 3-AP's in 2-coloring of integers. typed notes.
Lecture XXX: Roth Theorem, positive density subsets of [n] contains 3-AP, via Fourier method. typed notes.
Lecture XXXII: Solymosi: Wickets in 3-uniform hypergraphs.
Lecture XXXI: Number of k-SAT functions, rooted/weighted Hypergraph Turán Theorem.
Lecture XXXIII: Number of k-SAT functions, rooted/weighted Hypergraph Turán Theorem: explanation of the reduction step.
