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Explore the fundamentals of arithmetic Ramsey theory in this graduate-level lecture from the IAS/PCMI Park City Mathematics Institute. Begin with classical results proven through color focusing arguments, including van der Waerden's theorem which guarantees monochromatic k-term arithmetic progressions in any finite coloring of the integers. Progress to modern tools from additive combinatorics and learn to apply them in proving that various linear and nonlinear configurations are density or partition regular, with emphasis on achieving good quantitative bounds. Examine Ramsey-theoretic questions concerning arithmetic configurations in abelian groups as part of a comprehensive course on extremal and probabilistic combinatorics. Benefit from lecture notes and problem sets provided through the PCMI program, designed for graduate students with some background in Fourier analysis, particularly on finite abelian groups. This first part of the series establishes foundational concepts before advancing to more sophisticated techniques in subsequent lectures.
