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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 where a family of sets is called 'intersecting' if no two sets are disjoint, a principle that plays a 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 advanced 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 discover the numerous connections between intersecting families and other mathematical areas. Learn how this theory intersects with analysis, geometry, number theory, statistical physics, and theoretical computer science within the broader context of extremal and probabilistic combinatorics. Access accompanying lecture notes and problem sets to reinforce understanding of these sophisticated combinatorial structures and their applications across multiple mathematical disciplines.
