## Combinatorial Set Theory: Partition Relations for CardinalsThis 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

9 | |

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 |

341 | |

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### Common terms and phrases

according arbitrary assertion assumption Axiom of Choice cf(K claim coﬁnal coloring f combinatorial completes the proof contradicts Corollary countable deﬁned deﬁnition denotes disjoint sets easy elements equivalence classes established fact ﬁnite ﬁrst follows formula free set free with respect function f given Hajnal Hence homogeneous set implies inaccessible cardinal increasing sequence induction inequality inﬁnite cardinal integer Lemma Let f measurable cardinal nontrivial Note obtain ofthe order type ordered set ordinal ordinary partition relation pairwise disjoint partial order partition symbol partition tree preceding lemma prime ideal proof is complete prove Ramsey’s theorem regressive function regular cardinal replacing respect to f result right-hand side satisﬁed satisfying says sentence sequence of cardinals set mapping set of cardinality set of color set theory stationary set Stepping-up Lemma subset successor successor cardinal Theorem transﬁnite recursion uncountable veriﬁes verify wellordering write