Rob Morris: The method of hypergraph containers
In this short course we will discuss a recently-developed technique, known as the "method of hypergraph containers", that has proved useful in the study of extremal and Ramsey-type questions about sparse random objects. This general technique can be applied in a wide range of settings, and is based on the following fundamental fact: the independent sets in many 'natural' hypergraphs exhibit a certain kind of 'clustering', which allows one to count them one cluster at a time, using (in each case) a suitable 'supersaturation' theorem. We will attempt to motivate this abstract statement, sketch its proof, and give various applications of the method. In particular, we will discuss the threshold for the event that every r-colouring of the edges of G(n,p) contains a monochromatic copy of H.