In this thesis, we explore the method of hypergraph containers and its applications to problems in extremal combinatorics and discrete geometry. We start by introducing containers on triangle-free graphs and use it to give a lower bound on the number of triangle-free graphs on n vertices. We then generalise to the container algorithm for independent sets in graphs, which is a version of regular graphs. The proof of this theorem is an algorithm. We study this algorithm and apply it to the Erdős-Rényi random graph G(n,p). After this we expand to hypergraphs, specifically 3-uniform hypergraphs. Finally we look at the application to a problem in discrete geometry: Given n points in the Euclidean plane ℝ², with at most three on any line, how large a subset are we guaranteed to find in general position (i.e., with at most two on any line)? A strong upper bound is obtained by using the hypergraph container method.

In this text, we begin by giving a definition of Vector Space Ramsey Numbers. It concerns colourings of $t$-dimensional subspaces of some vector space $\mathbb{F}_q^n$. We want to ensure that each colouring contains a monochromatic $k$-dimensional subspace. After proving that these numbers always exist, we continue with studying asymptotic bounds for these numbers. We study a selection of methods, such as through coding theory or using the probabilistic method, to obtain lower and upper bounds for vector space Ramsey numbers. Lastly, we introduce two methods to directly compute vector space Ramsey numbers. That being through an ILP formulation and through a SAT formulation.