3 hours of lectures per week.
Lecturer
Jørgen Brandt
Content
We will look further into a number of topics introduced in part I. Transversal theory will be generalized to Matroid theory. Matroids encapsulate such fundamental concepts as linear independence in vector spaces, spanning trees in graphs and transversal set systems from pure combinatorics. The fact that not all matroids are created equal leads to deep (and far from solved) problems concerning their representability as vectorial matroids, as graphic matroids etc. Starting from the basic axioms we introduce matroid minors which are crucial for the above questions. This will lead to a characterization of binary matroids among other things.
This states that the only obstruction to binary representability is the four point line U2,4. Another large area is Ramsey theory. This includes Hales-Jewett's theorem (generalizing van der Waerden's famous theorem on arithmetic progressions in partitions of the natural numbers), Rado's complete characterization of so-called regular systems of equations (a generalization of Schur's result on sumfree sets), and Euclidian Ramsey theory where partitions of Euclidian spaces are studied. One of the fundamental problems in Design theory is the search for sufficient conditions for existence of designs with given parameters. A beautiful result is Wilson's asymptotic existence theory.
Prerequisites
Combinatorics I.
Literature
J. Oxley: Matroid Theory, Oxford 1992, and notes.