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Explore the fundamental theory of intersecting families in combinatorics through this graduate-level lecture from the 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 explore 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 areas of mathematics, with no prior knowledge of the field required. This lecture forms part of the PCMI 2025 Graduate Summer School focused on Probabilistic and Extremal Combinatorics, providing an in-depth introduction to preeminent themes and methods in these rapidly growing fields of discrete mathematics.
