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Explore advanced concepts in arithmetic Ramsey theory through this 59-minute graduate-level lecture presented by Sarah Peluse from Stanford University at the IAS/PCMI Park City Mathematics Institute. Delve into Ramsey-theoretic questions concerning arithmetic configurations in abelian groups, building upon classical results like van der Waerden's theorem that guarantees monochromatic k-term arithmetic progressions in any finite coloring of the integers. Learn how color focusing arguments work in proving these foundational results before advancing to modern tools from additive combinatorics. Discover how these contemporary techniques can be applied to prove that various linear and nonlinear configurations are density or partition regular, with emphasis on obtaining good quantitative bounds. Master the application of sophisticated mathematical tools to understand when certain patterns must appear in colored or structured sets. This lecture forms part of the PCMI 2025 Graduate Summer School focused on Probabilistic and Extremal Combinatorics, providing essential background for understanding connections between combinatorics, analysis, geometry, number theory, statistical physics, and theoretical computer science. Note that some background in Fourier analysis, particularly on finite abelian groups, would be beneficial for full comprehension of the material presented.
