Ramsey Theory on the Integers
Ramsey theory is the study of the structure of mathematical objects that is preserved under partitions. In its full generality, Ramsey theory is quite powerful, but can quickly become complicated. By limiting the focus of this book to Ramsey theory applied to the set of integers, the authors have produced a gentle, but meaningful, introduction to an important and enticing branch of modern mathematics. Ramsey Theory on the Integers offers students something quite rare for a book at this level: a glimpse into the world of mathematical research and the opportunity to begin pondering unsolved problems themselves. In addition to being the first truly accessible book on Ramsey theory, this innovative book also provides the first cohesive study of Ramsey theory on the integers. It contains perhaps the most substantial account of solved and unsolved problems in this blossoming subarea of Ramsey theory. The result is a breakthrough book that will engage students, teachers, and researchers alike.
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2-coloring of 1,n 3-term arithmetic progression admits a monochromatic assume avoids monochromatic b)-triples Chapter complete graph complete the proof consider contradiction Corollary Deﬁne defined deﬁnition denote der Waerden’s theorem descending wave elements equation Erd˝os Example finite ﬁrst g i g Hence hypergraph implies induction k-term monochromatic known least positive integer Lemma lower bound Math matic member of F metic monochro monochromatic 3-term arithmetic monochromatic arithmetic progression monochromatic k-term monochromatic member monochromatic Schur triples monochromatic set monochromatic solution monochromatic triangle monotone arithmetic progression notation Note pairs permutations pigeonhole principle polynomial progression of length progression with gap proof of Theorem prove quasi-progression of diameter r-coloring of Z+ r-large r-regular R(AD Rado Ramsey numbers Ramsey theory Ramsey-type function Ramsey’s theorem reader as Exercise References result Richard Rado Schur numbers Schur’s theorem semi-progression of scope Subcase subset Thue-Morse sequence upper bound values van der Waerden’s Waerden numbers Waerden’s theorem