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Explore advanced concepts in arithmetic Ramsey theory through this third lecture in a series 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 to apply modern tools from additive combinatorics to prove that various linear and nonlinear configurations are density or partition regular, with emphasis on achieving good quantitative bounds. Discover how color focusing arguments work in practice and gain exposure to sophisticated techniques that bridge combinatorics with other mathematical areas. This lecture assumes some background in Fourier analysis, particularly on finite abelian groups, and forms part of a comprehensive graduate program focusing on probabilistic and extremal combinatorics with connections to analysis, geometry, number theory, statistical physics, and theoretical computer science.
