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Explore the fourth part of a comprehensive lecture series on intersecting families in combinatorics, delivered by Imre Leader from the University of Cambridge at the IAS/PCMI Park City Mathematics Institute. Delve into the rich theory of intersecting families - collections of sets where no two are disjoint - which plays a central role in modern combinatorics. Discover fundamental theorems including the Erdős-Ko-Rado theorem, the second Erdős-Ko-Rado theorem, Katona's t-intersecting theorem, and the Ahlswede-Khachatrian theorem. Examine advanced topics such as odd and even intersections through the Frankl-Wilson theorem, applications to Borsuk's conjecture, coverings by intersecting families, uniform covers, triangle-intersecting families, and Alon's common graphs conjecture. Learn how intersecting families connect to various mathematical areas including analysis, geometry, number theory, statistical physics, and theoretical computer science. This lecture assumes no prior knowledge of the area and is part of the PCMI 2025 Research Topic on Probabilistic and Extremal Combinatorics, providing an in-depth introduction to preeminent themes and methods in these rapidly growing fields of discrete mathematics.
