In this minicourse we will describe a recent powerful technique in enumerative combinatorics: the container method. Informally, the container method says that hypergraphs satisfying certain natural conditions will have few independent sets. As many combinatorial problems can be posed as questions about hypergraphs, this simple notion has wide-ranging applications. A typical application is to count the number of finite objects with forbidden substructures. For example, counting the number of n-vertex C4-free graphs or counting the number of Sidon sets in [n].
Hypergraph container theorems were proved by Balogh, Morris and Samotij and, independently, by Saxton and Thomason. This work was quickly recognized as a significant breakthrough in combinatorics. Indeed, the five authors were awarded the George Pólya Prize in Combinatorics 2016 for their efforts.
In this series of lectures we will give a gentle introduction to this method. We will prove several container theorems and give as many nice applications as time allows.
Notes by Balogh Jozsi.
When: 2020 February 3-7, each day from 10:30-12.
Where: ELTE Déli Tömb 3-219
Credits: ELTE students can take the course in Neptun to receive credits.