Asymptotic Enumeration via Graph Containers and Entropy - Part 5

Explore advanced asymptotic enumeration techniques through specialized graph containers and entropy methods in this fifth lecture from the PCMI 2025 Graduate Summer School series. Learn about Sapozhenko's graph container methods specifically designed for independent sets in expander graphs, and discover how Shannon's entropy from information theory serves as a powerful tool for measuring expected information content in random variables. Examine both classical and contemporary applications of graph containers and entropy methods in solving various enumeration problems, with particular focus on their combined use in extremal and probabilistic combinatorics. Master the container methods that have revolutionized the field by providing powerful bounds for counting independent sets in graphs and hypergraphs, while understanding their profound influence on extremal and probabilistic combinatorics research. Gain insights into the intersection of discrete mathematics with analysis, geometry, number theory, statistical physics, and theoretical computer science through these sophisticated enumeration techniques. The lecture assumes introductory-level knowledge of combinatorics and probability, making advanced concepts accessible while maintaining mathematical rigor essential for graduate-level study in probabilistic and extremal combinatorics.