In this advanced-level course we will study some advanced applications of probabilistic methods in combinatorics and combinatorial number theory, including:
- The Erdös–Rényi random graph (chromatic number, threshold for spanning subgraphs).
- The Lovász Local Lemma, and applications.
- Dependent random choice, and applications in extremal graph theory, Ramsey theory, and additive combinatorics.
- Applications of quasirandomness in Ramsey theory.
- Martingales and concentration inequalities.
- Random graph processes (the Rödl nibble, the differential equations method), and applications to number theory.
- Discrepancy (six standard deviations suffice, the Beck–Fiala theorem).
- The method of hypergraph containers, and applications to random graphs.
 N. Alon and J.H. Spencer, The Probabilistic Method, 4th edition, Wiley, New York, 2016.
 J. Balogh, R. Morris and W. Samotij, The method of hypergraph containers, Proc. Int. Cong. Math., Rio de Janeiro, 2018, Vol. 3, 3045–3078.
 D. Conlon, J. Fox and B. Sudakov, Recent developments in graph Ramsey theory, Surveys in Combinatorics, Cambridge University Press, 2015.
 J. Fox and B. Sudakov, Dependent random choice, Random Structures Algorithms, 38 (2011), 1–32.