Издательство North-Holland, 1984, -342 pp.
Ramsey's classical theorem in its simplest form, published in 1930, says that if we put the edges of an infinite complete graph into two classes, then there will be an infinite complete subgraph all edges of which belong to the same class. The partition calculus developed as a collection of generalizations of this theorem. The first important generalization was the Erdos-Dushnik-Miller theorem which says that for an arbitrary infinite cardinal k, if we put the edges of a complete graph of cardinality k into two classes then either the first class contains a complete graph of cardinality k or the second one contains an infinite complete graph. An earlier result of Sierpinski says that in case k = 2No we cannot expect that either of the classes contains an uncountable complete graph. The first work of a major scope, which sets out to give a 'calculus' of partitions as its aim, was the paper "A partition calculus in set theory" written by Erdos and Rado in 1956. In 1965, Erdos, Hajnal, and Rado gave an almost complete discussion of the ordinary partition relation for cardinals under the assumption of the generalized continuum hypothesis. At that time there were hardly any general results for ordinals, though there were some results of Specker, and the paper of Erdos and Rado quoted above also contains some results for them. The situation has now changed considerably. The advent of Cohen's forcing method, and later Jensen's theory of the constructible universe gave new spurs to the development of the partition calculus. The main contributors in the next ten years were J. E. Baumgartner, C. C. Chang, F. Galvin, J. Larson, R. A. Laver, E. C. Milner, K. Prikry, and S. Shelah, to mention but a few. Independence results are beyond the scope of this book, though it will occasionally be useful to quote some of them in order to put theorems of set theory into their real perspective. An attempt to give a survey that deals also with independence results was made by Erdos and Hajnal in their paper "Solved and unsolved problems in set theory" which appeared in the Tarski symposium volume in 1974. The progress here is, however, so rapid that this survey was obsolete to a certain extent already when it appeared in print.
Our aim in writing this book is to present what we consider to be the most important combinatorial ideas in the partition calculus, and we also want to give a discussion of the ordinary partition relation for cardinals without the assumption of the generalized continuum hypothesis; we tried to make this latter as complete as possible. A separate section 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 the continuum, several sections on set mappings, and an account of recent inequalities for cardinal powers that were obtained in the wake of Silver's breakthrough result saying that the continuum hypothesis cannot first fail at a singular cardinal of uncountable cofinality. Large cardinals are discussed up to measurability, in slightly more detail than would be necessary strictly from the viewpoint of the partition calculus.
We assume some acquaintance with set theory on the part of the reader, though we tried to keep this to a minimum by the inclusion of an introductory chapter. The nature of the subject matter made it inevitable that we make some demands on the reader in the way of mathematical maturity. And we make another important assumption: the axiom of choice, that is the axiomatic framework in this book is Zermelo-Fraenkel set theory always with the axiom of choice. There are interesting results in the partition calculus which do not need the axiom of choice, but we have never made an attempt to avoid using it. There are many interesting assertions that are consistent with set theory without the axiom of choice but contradict this latter, and there are many important theorems of set theory plus some interesting additional assumption, e.g. the axiom of determinacy, that is known to contradict the axiom of choice. We did not include any of these; unfortunate though this may be, we had to compromise; we attempted to discuss infinity, but had to accomplish our task in finite time.
Introduction.
Preliminaries.
Fundamentals about Partition Relations.
Trees and Positive Ordinary Partition Relations.
Negative Ordinary Partition Relations, and the Discussion of the Finite Case.
The Canonization Lemmas.
Large Cardinals.
Discussion of the Ordinary Partition Relation with Superscript 2.
Discussion of the Ordinary Partition Relation with Superscript 3.
Some Applications OF Combinatorial Methods.
A Brief Survey of the Square Bracket Relation.
Ramsey's classical theorem in its simplest form, published in 1930, says that if we put the edges of an infinite complete graph into two classes, then there will be an infinite complete subgraph all edges of which belong to the same class. The partition calculus developed as a collection of generalizations of this theorem. The first important generalization was the Erdos-Dushnik-Miller theorem which says that for an arbitrary infinite cardinal k, if we put the edges of a complete graph of cardinality k into two classes then either the first class contains a complete graph of cardinality k or the second one contains an infinite complete graph. An earlier result of Sierpinski says that in case k = 2No we cannot expect that either of the classes contains an uncountable complete graph. The first work of a major scope, which sets out to give a 'calculus' of partitions as its aim, was the paper "A partition calculus in set theory" written by Erdos and Rado in 1956. In 1965, Erdos, Hajnal, and Rado gave an almost complete discussion of the ordinary partition relation for cardinals under the assumption of the generalized continuum hypothesis. At that time there were hardly any general results for ordinals, though there were some results of Specker, and the paper of Erdos and Rado quoted above also contains some results for them. The situation has now changed considerably. The advent of Cohen's forcing method, and later Jensen's theory of the constructible universe gave new spurs to the development of the partition calculus. The main contributors in the next ten years were J. E. Baumgartner, C. C. Chang, F. Galvin, J. Larson, R. A. Laver, E. C. Milner, K. Prikry, and S. Shelah, to mention but a few. Independence results are beyond the scope of this book, though it will occasionally be useful to quote some of them in order to put theorems of set theory into their real perspective. An attempt to give a survey that deals also with independence results was made by Erdos and Hajnal in their paper "Solved and unsolved problems in set theory" which appeared in the Tarski symposium volume in 1974. The progress here is, however, so rapid that this survey was obsolete to a certain extent already when it appeared in print.
Our aim in writing this book is to present what we consider to be the most important combinatorial ideas in the partition calculus, and we also want to give a discussion of the ordinary partition relation for cardinals without the assumption of the generalized continuum hypothesis; we tried to make this latter as complete as possible. A separate section 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 the continuum, several sections on set mappings, and an account of recent inequalities for cardinal powers that were obtained in the wake of Silver's breakthrough result saying that the continuum hypothesis cannot first fail at a singular cardinal of uncountable cofinality. Large cardinals are discussed up to measurability, in slightly more detail than would be necessary strictly from the viewpoint of the partition calculus.
We assume some acquaintance with set theory on the part of the reader, though we tried to keep this to a minimum by the inclusion of an introductory chapter. The nature of the subject matter made it inevitable that we make some demands on the reader in the way of mathematical maturity. And we make another important assumption: the axiom of choice, that is the axiomatic framework in this book is Zermelo-Fraenkel set theory always with the axiom of choice. There are interesting results in the partition calculus which do not need the axiom of choice, but we have never made an attempt to avoid using it. There are many interesting assertions that are consistent with set theory without the axiom of choice but contradict this latter, and there are many important theorems of set theory plus some interesting additional assumption, e.g. the axiom of determinacy, that is known to contradict the axiom of choice. We did not include any of these; unfortunate though this may be, we had to compromise; we attempted to discuss infinity, but had to accomplish our task in finite time.
Introduction.
Preliminaries.
Fundamentals about Partition Relations.
Trees and Positive Ordinary Partition Relations.
Negative Ordinary Partition Relations, and the Discussion of the Finite Case.
The Canonization Lemmas.
Large Cardinals.
Discussion of the Ordinary Partition Relation with Superscript 2.
Discussion of the Ordinary Partition Relation with Superscript 3.
Some Applications OF Combinatorial Methods.
A Brief Survey of the Square Bracket Relation.