Ramsey Theory and Random Graphs

This advanced course will cover fundamental results and recent developments in the areas of Ramsey theory and random graphs, including:

1. The Probabilistic Method: the Lovász Local Lemma, the Alon–Rödl method, dependent random choice, multicolour Ramsey numbers, the H-free process.
2. Graph Ramsey theory: upper bounds on R (K, L), hypergraph Ramsey numbers, Ramsey numbers of sparse graphs: the Graham–Rödl–Rucinski method, Ramsey numbers of degenerate graphs, Ramsey goodness, size Ramsey numbers.
3. Pseudorandom graphs and applications to Ramsey numbers.
4. Hypergraph containers and applications to Ramsey problems in random graphs., in spare time, and its application to moduli.

[1] N. Alon and J. H. Spencer, The Probabilistic Method, 4th edition, Wiley, 2016.
[2] J. Balogh, R. Morris, W. Samotij, The method of hypergraph containers, ICM 2018.
[3] F. Botler, M. Collares, T. Martins, W. Mendonça, R. Morris and G. Mota, Combinatória, XXXIII Colóquio Brasileiro de Matemática (SBM/IMPA).
[4] D. Conlon, Pseudorandom graphs, available at http://www.its.caltech.edu/ dconlon/PSlectures.pdf
[5] D. Conlon, J. Fox and B. Sudakov, Recent developments in graph Ramsey theory, In: Surveys in Combinatorics, Cambridge University Press, 2015.
[6] J. Fox and B. Sudakov, Dependent random choice, Random Structures Algorithms, 38 (2011), 1–32.