Combinatorial Set Theory: Partition Relations for Cardinals

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Elsevier, Aug 18, 2011 - Mathematics - 348 pages
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This work presents the most important combinatorial ideas in partition calculus and discusses ordinary partition relations for cardinals without the assumption of the generalized continuum hypothesis. A separate section of the book describes the main partition symbols scattered in the literature. A chapter on the applications of the combinatorial methods in partition calculus includes a section on topology with Arhangel'skii's famous result that a first countable compact Hausdorff space has cardinality, at most continuum. Several sections on set mappings are included as well as an account of recent inequalities for cardinal powers that were obtained in the wake of Silver's breakthrough result saying that the continuum hypothesis can not first fail at a singular cardinal of uncountable cofinality.

 

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Contents

Chapter I Introduction
9
Chapter II Preliminaries
34
Chapter III Fundamentals about partition relations
52
Chapter IV Trees and positive ordinary partition relations
80
Chapter V Negative ordinary partition relations and the discussion of the finite case
105
Chapter VI The canonization lemmas
158
Chapter VII Large cardinals
168
Chapter VIII Discussion of the ordinary partition relation with superscript 2
215
Chapter IX Discussion of the ordinary partition relation with superscript G 3
233
Chapter X Some applications of combinatorial metbods
263
Chapter XI A brief survey of the square bracket relation
313
Bibliography
335
Author index
341
Subject index
343
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