Combinatorics using some algebraic constructions. We use dimension arguments to get bounds on interesting combinatorial numbers. We study the eigenvalues of adjacency matrices on graphs to get information about graphs at hand. This has great applications in the so called extremal combinatorics.

We will be studying spectral theory on graphs. This has some interesting combinatorial consequences on graph properties, like the maximum independent set and covering properties. The highlight of this section is Erdos-Ko-Rado theorem and the impossibility of covering K_10 with three copies of the Petersen graph.

In combinatorial geometry we will be studying combinatorial identities and inequalities that relate to point sets and polytopes. For instance, how many points in R^d can you find such that the distance between any two of them is one of two given real numbers? We will find bounds for these quantities using linear algebra.

Finally, we will be concluding the course with some polynomial constructions, applications of Chevalier - Warning theorem and the combinatorial Nullstellensatz. We will use these to prove that some graphs contain a three-regular graph, and to estimate the list-colouring number of a graph.

We will be following Linear algebra methods in combinatorics, by László Babai and Péter Frankl as well as a lecture series from 2015 from Benny Sudakov.

Linear algebra, algebra (determinants, eigenvalues).

Abstract algebra or module arithmetic