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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 that a family of sets is called 'intersecting' if no two sets are disjoint, and discover how this seemingly simple definition leads to profound mathematical insights that play 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 while examining their applications and connections to other mathematical areas. Investigate advanced topics such as odd and even intersections through the Frankl-Wilson theorem, explore applications to Borsuk's conjecture, and analyze coverings by intersecting families including uniform covers and triangle-intersecting families. Examine Alon's common graphs conjecture and discover the extensive links between intersecting families and other areas of mathematics including analysis, geometry, number theory, statistical physics, and theoretical computer science. Benefit from clear explanations that assume no prior knowledge of the area while building toward sophisticated understanding of extremal and probabilistic combinatorics, supported by accompanying lecture notes and problem sets from the PCMI 2025 program on Probabilistic and Extremal Combinatorics.
