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Explore the rich theory of intersecting families in combinatorics through this graduate-level lecture delivered by Imre Leader from the University of Cambridge at the IAS/PCMI Park City Mathematics Institute. Delve into the fundamental concept of intersecting families, where no two sets in the family are disjoint, and discover their central role in modern combinatorics. Master key theorems including the Erdős-Ko-Rado theorem, the second Erdős-Ko-Rado theorem, Katona's t-intersecting theorem, and the Ahlswede-Khachatrian theorem. Examine specialized topics such as odd and even intersections through the Frankl-Wilson theorem, applications to Borsuk's conjecture, coverings by intersecting families, uniform covers, and triangle-intersecting families. Investigate Alon's common graphs conjecture and understand the connections between intersecting families and other mathematical areas. Learn how these concepts apply to extremal and probabilistic combinatorics, with emphasis on their connections to 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 structures and their applications across multiple mathematical disciplines.
