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Explore the third part of a graduate-level lecture series on extremal graph theory delivered by David Conlon from Caltech at the IAS/PCMI Park City Mathematics Institute. Delve into the fundamental concept of extremal numbers ex(n, H), which represents the maximum number of edges possible in an H-free graph with n vertices. Examine the current understanding of this function for non-bipartite graphs while focusing on the abundance of open questions that remain for bipartite cases. Learn about recent progress and breakthrough developments in extremal graph theory for bipartite graphs, a central area of contemporary discrete mathematics. Discover how extremal combinatorics investigates the maximum or minimum size of discrete structures under specific restrictions, and understand its connections to other mathematical fields including analysis, geometry, number theory, statistical physics, and theoretical computer science. Access accompanying lecture notes and problem sets to reinforce the theoretical concepts presented in this comprehensive mathematical exploration of one of the most active research areas in modern combinatorics.
