Combinatorial Set Theory: Partition Relations for Cardinals (eBook)

Front Cover 1
Combinatorial Set Theory: Partition Relations for Cardinals 4
Copyright Page 5
Contents 8
Preface 6
Chapter I. Introduction 10
1. Notation and basic concepts 10
2. The axioms of Zermelo–Fraenkel set theory 14
3. Ordinals cardinals, and order types 17
4. Basic tools of set theory 22
Chapter II. Preliminaries 35
5. Stationary sets 35
6. Equalities and inequalities for cardinals 40
7. The logarithm operation 46
Chapter III. Fundamentals about partition relations 53
8. A guide to partition symbols 53
9. Elementary properties of the ordinary partition symbol 63
10. Ramsey’s theorem 66
11. The Erdös–Dushnik–Miller theorem 71
12. Negative relations with infinite superscripts 79
Chapter IV. Trees and positive ordinary partition relations 81
13.Trees 81
14. Tree arguments 83
15. End-homogeneous sets 87
16. The Stepping-up Lemma 91
17. The main results in case r = 2 and k is regular and some corollaries for r G 3
18. A direct construction of the canonical partition tree 101
Chapter V. Negative ordinary partition relations, and the discussion of the finite case 106
19. Multiplication of negative partition relations for r = 2 106
20. A negative partition relation established with the aid of GCH 111
21. Addition of negative partition relations for r =2 114
22. Addition of negative partition relations for r G 3 118
23. Multiplication of negative partition relations in case r G 3 123
24. The Negative Stepping-up Lemma 133
25. Some special negative partition relations for r G 3 139
26. The finite case of the ordinary partition relation 146
Chapter VI. The canonization lemmas 159
27. Shelah's canonization 159
28. The General Canonization Lemma 164
Chapter VII. Large cardinals 169
29. The ordinary partition relation for inaccessible cardinals 169
30. Weak compactness and a metamathematical approach to the Hanf–Tarski result 178
31. Baumgartner's principle 189
32. A combinatorial approach to the Hanf-Tarski result 195
33. Hanf's iteration scheme 201
34. Saturated ideals, measurable cardinals. and strong partition relations 202
Chapter VIII. Discussion of the ordinary partition relation with superscript 2 216
35. Discussion of the ordinary partition symbol in case r = 2 216
36. Discussion of the ordinary partition relation in case r=2 under the assumption of GCH 227
37. Sierpinski partitions 230
Chapter IX. Discussion of the ordinary partition relation with superscript G 3 234
38. Reduction of the superscript 234
39. Applicability of the Reduction Theorem 241
40. Consequences of the Reduction Theorem 244
41. The main result for the case r G 3 254
42. The main result for the case r G 3 with GCH 260
Chapter X. Some applications of combinatorial metbods 264
43. Applications in topology 264
44. Fodor's and Hajnal's set-mapping theorems 273
45. Set mapping of type > 1
46. Finite free sets with respect to set mappings of type > 1
47. Inequalities for powers of singular cardinals 289
48. Cardinal exponentiation and saturated ideals 302
Chapter XI. A brief survey of the square bracket relation 314
49. Negative square bracket relations and the GCH 314
50. The effect of a Suslin tree on negative relations 319
51. The Kurepa Hypothesis and negative stepping up to superscript 3 322
52. Aronszajn trees and Specker types (preparations for results without GCH) 325
53. Negative square bracket relations without GCH 328
54. Positive relations for singular strong limit cardinals 331
55. Infinitary Jónsson algebras 333
Bibliography 336
Author index 342
Subject index 344