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Explore the fundamentals of extremal graph theory in this 51-minute lecture from the IAS/PCMI Park City Mathematics Institute. Learn about the extremal number ex(n, H), which represents the maximum number of edges in an H-free graph with n vertices, and discover why this function remains well understood for non-bipartite graphs while presenting numerous open questions for bipartite cases. Examine recent progress made in understanding bipartite extremal problems through detailed mathematical analysis and proofs. Delve into how extremal combinatorics studies the maximum or minimum size of discrete structures under specific restrictions, connecting to broader themes in analysis, geometry, number theory, statistical physics, and theoretical computer science. Access accompanying lecture notes and problem sets to reinforce your understanding of these advanced combinatorial concepts as part of the comprehensive graduate program on probabilistic and extremal combinatorics.
