Asymptotic Enumeration via Graph Containers and Entropy - Part 1

Explore asymptotic enumeration techniques through graph containers and entropy methods in this 51-minute lecture from the IAS/Park City Mathematics Institute. Learn about specialized graph containers developed by Sapozhenko (1987) for independent sets in expanders, and discover how Shannon's entropy from information theory applies to enumeration problems. Examine the container methods as powerful tools for bounding independent sets in graphs and hypergraphs, understanding their significant influence in extremal and probabilistic combinatorics. Delve into both classical and contemporary applications of graph containers, entropy methods, and their combinations across various enumeration challenges. Master the fundamental connections between these mathematical concepts while building upon introductory-level knowledge of combinatorics and probability. This lecture forms part of a comprehensive series on probabilistic and extremal combinatorics, two central branches of discrete mathematics that study the size constraints of discrete structures and investigate random combinatorial objects using probability theory and combinatorial methods.