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Explore the fascinating world of Ramsey numbers in this 56-minute conference talk delivered at the IAS/PCMI Park City Mathematics Institute. Begin with a gentle introduction to Ramsey numbers designed for newcomers to the field, then delve into cutting-edge research on upper bounds for multicolour Ramsey numbers R_r(k). Discover how this fundamental area of combinatorics connects graph theory, probability, and extremal mathematics through accessible explanations followed by advanced mathematical insights. Learn about the historical development of Ramsey theory and understand why these problems remain among the most challenging and important questions in discrete mathematics. Gain exposure to recent breakthrough techniques and methods used to establish improved upper bounds, providing both foundational knowledge and current research perspectives in this active area of mathematical investigation.
