Logic Seminar: Nonstandard analysis in arithmetic Ramsey Theory

Arithmetic Ramsey theory is an area of combinatorics that focuses on the existence, for any finite coloring (partition)of the natural numbers, of “monochromatic patterns” defined by arithmetic operations. A classical result is van der Waerden's Theorem, which states that monochromatic arithmetic progressions of any prescribed (finite) length can always be found. A fundamental result on infinite configurations is Hindman's Theorem, which states that one can always find an infinite sequence such that all sums of its distinct elements are monochromatic. Various techniques have been successfully applied to the problems of arithmetic Ramsey theory, including ergodic theory, discrete harmonic analysis, topological dynamics, algebra in the space of ultrafilters. In this seminar I will show some recent applications in this area of nonstandard analysis methods combined with the use of ultrafilters.