Instructor:
Lecture Outline:
Lecture 1 Definitions on Turan problems Lecture 2 Kovari-Sos-Turan Theorem, Turan density for blowups of complete k-partite k-graphs. Lecture 3 Supersaturation Lemma, Turan density on blowups Lecture 4 Kovari-Sos-Turan Theorem for Hypergraphs, Counting the number of blowups of complete k-partite k-graphs. Lecture 5 Turan's Theorem, Erdos-Stone-Simonovits Theorem Lecture 6 A quantitative bound of Erdos-Stone-Simonovits Theorem, Moon-Moser inequalities Lecture 7 An upper bound on Turan numbers of complete hypergraphs (De-Caen's Theorem) Lecture 8 A lower bound on Turan numbers of complete hypergraphs Lecture 9 Simonovits' stability Theorem, Furedi's new proof on the stability of ex(n,K_r) Lecture 10 Turan numbers of edge-critical graphs - an application of the stability method, definition of decomposition family Lecture 11 Andrasfai-Erdos-Sos Theorem, Chromatic threshold Lecture 12 Erdos-Sos conjecture on the Turan numbers of trees, Path lengths in graphs, Posa's rotation, Closure, Hamiltonian properties. Lecture 13 Cycle lengths in 2-connected graphs, A theorem of Erdos-Gallai on cycles. Lecture 14 Counting the number of walks. Lecture 15 Counting the number of non-backtracting walks, and Moore Bound. Lecture 16-18 Hoffman-Singleton Theorem, Bondy-Simonovits Theorem on the upper bound of ex(n,C_{2k}). Lecture 19 Number of consecutive even cycles in graphs with large average degree and large girth; dependent random choice I Lecture 20 Dependent random choice II Lecture 21 Dependent random choice III; the general bounds on Turan numbers of bipartite graphs. Lecture 22-23 Szemeredi's regularity lemma