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Soft cover. Zustand: New. 8vo (21.5 cm), 70 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 2. A concise research monograph on the application of ultrafilter methods to the combinatorics of numbers, by Igor Protasov of the National University of Kyiv, a specialist in topological algebra. Original…ly prepared as lecture notes for a course at the Faculty of Mechanics and Mathematics of Kyiv University, the text was translated into English by Michael Zarichnyi and Taras Banakh for this edition. The book presents a self-contained introduction to the natural semigroup structure of the Stone-Čech compactification βℕ of the discrete space of positive integers--a construction which, from the late 1970s onward, became one of the central tools of modern Ramsey theory and topological algebra through the work of Hindman, van Douwen, Pym, Blass, Bergelson, Strauss, and others. An opening historical survey reviews the classical foundations of the subject, including Schur's theorem (1916), van der Waerden's theorem on arithmetic progressions (1927), Ramsey's theorem (1930), and Rado's extension to systems of linear homogeneous Diophantine equations (1933). The eleven subsequent sections develop the theory of filters and ultrafilters, ultrafilters on topological spaces, and the semigroup structure of βℕ, before presenting ultrafilter proofs of the theorems of Ramsey, Hindman, van der Waerden, Hales-Jewett, Rado, and Furstenberg-Weiss. The concluding section, on partitions of groups and rings, incorporates the author's own results published in 1993. A compact but influential introduction to one of the most powerful modern methods in Ramsey theory and topological algebra, intended for researchers and graduate students in combinatorics, algebra, and topology.
