Georgia Institute of Technology College of Sciences

Many combinatorial questions can be formulated as problems about independent sets in uniform hypegraphs, including questions about number of sets with no $k$-term arithmetic progression and questions about typical structure of $H$-free graphs. Balogh, Morris, and Samotij and, independently, Saxton and Thomason gave an approximate structural characterization of all independent sets in uniform hypergraphs with natural contitions on edge distributions, using something called "containers". We will go through the proof of the hypergraph container result of Balogh, Morris, and Samotij. We will also discuss some applications of this container result.