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Explore the fundamental concepts and recent advances in Turán numbers of hypergraphs through this comprehensive 2-hour seminar from the Institute for Advanced Study's Computer Science/Discrete Mathematics series. Begin with a survey of foundational results in extremal combinatorics, focusing on the classical Turán problem that seeks to determine the maximum number of edges in a forbidden substructure-free hypergraph given a fixed number of vertices. Learn essential techniques including Lagrangians and supersaturation methods that form the backbone of modern extremal graph theory. Delve into cutting-edge research on Turán numbers of long tight cycles, which represent a natural generalization of cycles to hypergraphs, and discover how these structures connect to broader questions in combinatorial optimization. Examine a novel hypergraph analogue of the fundamental graph theory result characterizing bipartite graphs through the absence of odd closed walks, extending this concept to r-uniform hypergraphs through colorings of (r-1)-tuples of vertices. Understand how this theoretical framework provides a unified approach to previously disparate results in uniformity r=3 by researchers including Kamčev, Letzter, Pokrovskiy, Balogh, and Luo, while opening pathways to analyze Turán numbers for much broader families of cycle-like hypergraphs.
