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Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite. An exciting new direction for combinatorics, this book will interest graduate students and researchers working in mathematical subdisciplines requiring the mastery and practice of high-dimensional Ramsey theory.

Algebraic spaces. --- Ramsey theory. --- Ramsey theory --- Algebraic spaces --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Spaces, Algebraic --- Geometry, Algebraic --- Combinatorial analysis --- Graph theory --- Analytic set. --- Axiom of choice. --- Baire category theorem. --- Baire space. --- Banach space. --- Bijection. --- Binary relation. --- Boolean prime ideal theorem. --- Borel equivalence relation. --- Borel measure. --- Borel set. --- C0. --- Cantor cube. --- Cantor set. --- Cantor space. --- Cardinality. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Combinatorics. --- Compact space. --- Compactification (mathematics). --- Complete metric space. --- Completely metrizable space. --- Constructible universe. --- Continuous function (set theory). --- Continuous function. --- Corollary. --- Countable set. --- Counterexample. --- Decision problem. --- Dense set. --- Diagonalization. --- Dimension (vector space). --- Dimension. --- Discrete space. --- Disjoint sets. --- Dual space. --- Embedding. --- Equation. --- Equivalence relation. --- Existential quantification. --- Family of sets. --- Forcing (mathematics). --- Forcing (recursion theory). --- Gap theorem. --- Geometry. --- Ideal (ring theory). --- Infinite product. --- Lebesgue measure. --- Limit point. --- Lipschitz continuity. --- Mathematical induction. --- Mathematical problem. --- Mathematics. --- Metric space. --- Metrization theorem. --- Monotonic function. --- Natural number. --- Natural topology. --- Neighbourhood (mathematics). --- Null set. --- Open set. --- Order type. --- Partial function. --- Partially ordered set. --- Peano axioms. --- Point at infinity. --- Pointwise. --- Polish space. --- Probability measure. --- Product measure. --- Product topology. --- Property of Baire. --- Ramsey's theorem. --- Right inverse. --- Scalar multiplication. --- Schauder basis. --- Semigroup. --- Sequence. --- Sequential space. --- Set (mathematics). --- Set theory. --- Sperner family. --- Subsequence. --- Subset. --- Subspace topology. --- Support function. --- Symmetric difference. --- Theorem. --- Topological dynamics. --- Topological group. --- Topological space. --- Topology. --- Tree (data structure). --- Unit interval. --- Unit sphere. --- Variable (mathematics). --- Well-order. --- Zorn's lemma.

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The analysis of the characteristics of walks on ordinals is a powerful new technique for building mathematical structures, developed by the author over the last twenty years. This is the first book-length exposition of this method. Particular emphasis is placed on applications which are presented in a unified and comprehensive manner and which stretch across several areas of mathematics such as set theory, combinatorics, general topology, functional analysis, and general algebra. The intended audience for this book are graduate students and researchers working in these areas interested in mastering and applying these methods.

Number theory --- Discrete mathematics --- discrete wiskunde --- getallenleer

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This book introduces graduate students and resarchers to the study of the geometry of Banach spaces using combinatorial methods. The combinatorial, and in particular the Ramsey-theoretic, approach to Banach space theory is not new, it can be traced back as early as the 1970s. Its full appreciation, however, came only during the last decade or so, after some of the most important problems in Banach space theory were solved, such as, for example, the distortion problem, the unconditional basic sequence problem, and the homogeneous space problem. The book covers most of these advances, but one of its primary purposes is to discuss some of the recent advances that are not present in survey articles of these areas. We show, for example, how to introduce a conditional structure to a given Banach space under construction that allows us to essentially prescribe the corresponding space of non-strictly singular operators. We also apply the Nash-Williams theory of fronts and barriers in the study of Cezaro summability and unconditionality present in basic sequences inside a given Banach space. We further provide a detailed exposition of the block-Ramsey theory and its recent deep adjustments relevant to the Banach space theory due to Gowers.

Functional analysis --- Discrete mathematics --- discrete wiskunde --- functies (wiskunde)