Content: Some or all of the following topics:
- Partially ordered sets and set systems: Dilworth's theorem, Sperner's theorem, the LYM inequality, the Sauer-Shelah Lemma
- Symmetric functions, Young Tableaux
- Designs and codes: Latin squares, finite projective planes, error-correcting codes
- Colouring: the chromatic polynomial
- Geometric combinatorics: Caratheodory's Theorem, Helly's Theorem, Radon's Theorem
- Probabilistic method: the existence of graphs with large girth and high chromatic number, use of concentration bounds
- Matroid theory: basic concepts, Rado's Theorem
- Regularity method: regularity lemma without a proof, the existence of 3-APs in dense subsets of integers
To give the students an opportunity to learn some of the more advanced combinatorial methods, and to see combinatorics in a broader context of mathematics.
By the end of the module the student should be able to:
- State and prove particular results presented in the module
- Adapt the presented methods to other combinatorial settings
- Apply simple probabilistic and algebraic arguments to combinatorial problems
- Use presented discrete abstractions of geometric and linear algebra concepts
- Derive approximate results using the regularity method
R. Diestel: Graph Theory, Springer, 4th edition, 2012.
R. Stanley: Algebraic Combinatorics: Walks, Trees, Tableaux and More, Springer, 2013.