On the Container Method in Graph Theory and Combinatorics

In this survey talk, we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This powerful tool using probabilistic methods exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; that is, it finds a realatively small collection of well-behaved, sparse containers that cover all the interesting objects.  We attempt to convey a general high-level overview of the method and focus on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry all while avoiding technical details as much as possible.