• Partially order sets: Dilworth's theorem, Sperner's theorem
• Planar graphs: MacLane's criterion, Whitney's theorem, Five Color Theorem and its list coloring generalization
• Algebraic combinatorics: Symmetric functions, Young Tableaux, RSK correspondance
• Designs and codes: Latin squares, finite projective planes, self-correcting codes
• Geometric combinatorics: simplicial complexes, combinatorics of polytopes, relation to optimization
• Probabilistic method: the existence of high-girth graphs with high chromatic number, use of concentration bounds
• Matroid theory: basic concepts, relation to algorithms and optimization, matroid intersection 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.