In this seminar, we shall present and discuss a variety of recent developments in the (broadly understood) areas of extremal and probabilistic combinatorics that are not normally covered in the standard courses.

Students registering for the seminar are expected to possess working knowledge of basic graph theory notions and be familiar with basic concepts of discrete probability.

Below is a tentative list of topics to be discussed and presented during the seminar. Topics that have already been reserved are highlighted in light gray.

1. Size Ramsey numbers of paths
   A. Dudek, P. Prałat, An alternative proof of the linearity of the size-Ramsey number of paths, to appear in Combinatorics, Probability and Computing (arXiv)
2. Chromatic and homomorphism thresholds of cliques
   H. Oberkampf, M. Schacht, On the structure of dense graphs with fixed clique number, preprint (arXiv)
3. Monotone paths in edge-ordered graphs
   K. G. Milans, Monotone paths in dense edge-ordered graphs, preprint (arXiv)
4. The number of colors strongly affects some hypergraph Ramsey numbers
   D. Conlon, J. Fox, V. Rödl, Hedgehogs are not colour blind, preprint (arXiv)
5. Notions of hypergraph Ramsey numbers and their equivalences
   D. Mubayi, A. Suk, Off-diagonal hypergraph Ramsey numbers, preprint (arXiv)
6. Possible exponents of graph Turán numbers
   B. Bukh, D. Conlon, Rational exponents in extremal graph theory, preprint (arXiv)
7. Triangle-free pseudo-random graphs
   N. Alon, Explicit Ramsey graphs and orthonormal labelings, Electronic Journal of Combinatorics 1 (1994), R#12 (arXiv)
   D. Conlon, A sequence of triangle-free pseudorandom graphs, preprint (arXiv)
   
8. Shannon's entropy and counting (as preparation for the next topic)
   D. Galvin, Three tutorial lectures on entropy and counting, survey (arXiv)
9. The Erdős–Simonovits conjecture (aka Sidorenko's conjecture)
   D. Conlon, J. H. Kim, C. Lee, J. Lee, Some advances on Sidorenko's conjecture, preprint (arXiv)
   B. Szegedy, An information theoretic approach to Sidorenko's conjecture, preprint (arXiv)
   
10. The Erdős–Szekeres conjecture
    G. Vlachos, On a conjecture of Erdős and Szekeres, preprint (arXiv)
    S. Norin, Y. Yuditsky, Erdős–Szekeres without induction, preprint (arXiv)
    
11. Stability of the Erdős–Ko–Rado theorem
    S. Das, T. Tran, Removal and Stability for Erdős–Ko–Rado, preprint (arXiv)
12. Enumerating graphs without a cycle of length four
    D. J. Kleitman, K. J. Winston, On the number of graphs without 4-cycles, Discrete Math. 41 (1982), 167–172. (article)
13. An analogue of Turán's theorem in pseudo-random graphs
    B. Sudakov, T. Szabó, V. H. Vu, A generalization of Turán's theorem, J. Graph Theory 49 (2005), 187–195. (article)
14. Small infecting sets in the bootstrap percolation on the hypercube
    N. Morrison, J. A. Noel Extremal Bounds for Bootstrap Percolation in the Hypercube, preprint (arXiv)
15. Ramsey numbers of degenerate graphs
    C. Lee, Ramsey numbers of degenerate graphs, preprint (arXiv)
16. The Erdős discrepancy problem
    T. Tao, The Erdős discrepancy problem, to appear in Discrete Analysis (arXiv)