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Explore the second part of a graduate-level lecture 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 made in addressing these challenging problems in bipartite extremal graph theory. This lecture forms part of the PCMI 2025 Graduate Summer School on Probabilistic and Extremal Combinatorics, organized by leading researchers from LSE, Stanford University, University of Waterloo, IMPA, and Tel Aviv University. Access accompanying lecture notes and problem sets to deepen your understanding of this central branch of contemporary discrete mathematics that connects with analysis, geometry, number theory, statistical physics, and theoretical computer science.
