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Explore fundamental methods in modern combinatorics through this 58-minute lecture focusing on Ramsey theory in the context of graphs. Learn how this single topic has inspired numerous techniques used throughout combinatorics and beyond, as presented by Julian Sahasrabudhe from the University of Cambridge at the IAS/PCMI Park City Mathematics Institute. Discover the connections between extremal and probabilistic combinatorics, two central branches of contemporary discrete mathematics that study how large or small discrete structures can be under certain restrictions and investigate random combinatorial objects using probability theory. Gain insight into how these fields connect with other areas of mathematics including analysis, geometry, number theory, statistical physics, and theoretical computer science. This lecture serves as part of a comprehensive graduate program designed to provide in-depth introduction to preeminent themes and methods in extremal and probabilistic combinatorics, with accompanying lecture notes and problem sets available for further study.
