Asymptotic Enumeration via Graph Containers and Entropy - Part 3

Explore advanced asymptotic enumeration techniques through graph containers and entropy methods in this 51-minute lecture from the IAS/PCMI 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 powerful container methods used to bound independent sets in graphs and hypergraphs, which have significantly influenced extremal and probabilistic combinatorics. Understand the combination of graph containers and entropy methods for solving various enumeration problems, with applications spanning from classical results to contemporary research. Master the theoretical foundations that connect discrete mathematics with information theory, probability, and combinatorial optimization. Gain insights into how these methods contribute to the broader landscape of extremal and probabilistic combinatorics, including connections to analysis, geometry, number theory, statistical physics, and theoretical computer science. Requires introductory-level knowledge of combinatorics and probability theory.