1- Ramsey theory: van der Waerden, Rado and Hales-Jewett theorems; Ramsey numbers of graphs with bounded degree, quotas for diagonal, off-diagonal and hypergraph Ramsey numbers. Recent progress in Ramsey theory for graphs.
2. The probabilistic method: the giant component, the chromatic number of G(n,p), the local lemma, Rodl’s nibble, random graph processes, concentration inequalities, dependent random choice.
3. Algebraic and topological methods: the polynomial method (the combinatorial nullstellensatz, Kakeya’s conjecture in finite bodies); the Frankl-Wilson modular inequality and applications (refutation of Borsuk’s conjecture, constructive lower bounds for Ramsey numbers, the chromatic number of the plane); applications of the Borsuk-Ulam theorem.
4. Additive combinatorics: sum-sets (Cauchy-Davenport and Freiman theorems), Balog-Szemerédi-Gowers theorem, non-sum sets (Cameron-Erdos conjecture), Roth’s theorem, discussion of Szemerédi’s theorem.
5. The container method for hypergraphs: extremal and Ramsey problems in G(n,p), counting and typical structure of graphs without copies of H, Folkman numbers, induced Ramsey numbers, constructions in discrete geometry.

References:
[1] N. Alon and J. H. Spencer. – The probabilistic method, 3rd edition, Wiley, New York, 2008.
[2] L. Babai and P. Frankl. – Linear algebra methods in combinatorics, Department of Computer Science, University of Chicago, preliminary version, 1991.
[3] B. Bollobás. – Modern Graph Theory (Graduate Texts in Mathematics), Springer-Verlag, New York, 1998.
[4] J. Matousek. – Thirty-three miniatures (mathematical and algorithmic applications of linear algebra), AMS, 2010.
[5] T. Tao and V. Vu. – Additive combinatorics, Cambridge Studies in Advanced Mathematics, CUP, 2006.
* Basic syllabus. The teacher has the autonomy to make any changes.