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Explore the fundamental theory of intersecting families in combinatorics through this graduate-level lecture from the IAS/PCMI Park City Mathematics Institute. Learn about families of sets where no two sets are disjoint, a concept that plays a central role in modern combinatorics. Discover 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 odd and even intersections through the Frankl-Wilson theorem and see applications to Borsuk's conjecture. Investigate coverings by intersecting families, uniform covers, and triangle-intersecting families, concluding with Alon's common graphs conjecture. Gain insight into the rich connections between intersecting families and other mathematical topics, with no prior knowledge of the area required. Access accompanying lecture notes and problem sets to deepen your understanding of this central topic in extremal and probabilistic combinatorics.
