# MA3J2 Combinatorics II

Lecturer: Keith Ball

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 Lectures

Assessment: 100% Examination

Formal registration prerequisites: None

Assumed knowledge:

MA241 Combinatorics I:

• Graph theory
• Hall's Theorem
• Graph colouring
• Norms on Euclidean space
• Open and closed sets
• Compactness
• Basic examples of finite fields
• Events
• Probabilities
• Random variables

Useful background:

• Poisson distribution
• Chebyshev's inequality
• The Central Limit Theorem
• Projective geometry

Synergies: The following module goes well together with Combinatorics II:

MA3G7 Functional Analysis I

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

Aims:
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.

Objectives:
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

Books:
R. Diestel: Graph Theory, Springer, 4th edition, 2012.
R. Stanley: Algebraic Combinatorics: Walks, Trees, Tableaux and More, Springer, 2013.