1.1. Note.

Translated from the 1983 Hungarian original by Attila Máté.

1.2. From the Publisher.

This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required. Exercises are included, and the more difficult ones are supplied with hints. An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part. The final part gives an introduction to modern tools of combinatorial set theory. This part contains enough material for a graduate course of one or two semesters. The subjects discussed include stationary sets, delta systems, partition relations, set mappings, measurable and real-valued measurable cardinals. Two sections give an introduction to modern results on exponentiation of singular cardinals, and certain deeper aspects of the topics are developed in advanced problems.

1.3. From the Preface.

This textbook was prepared on the basis of courses and lectures by András Hajnal for mathematics majors at Roland Eötvös University in Budapest, Hungary. The first edition appeared in 1983 since then the book went through a number of new printings and editions. During each of these, new problems were added and the historical remarks were updated. A number of revisions have also been made in the present, the first English, edition. A significant one among these is that hints were added for the problems in Part II, and a completely new section (Section 20) discusses the so-called square-bracket symbol.

The book consists of two parts and an Appendix to Part I. The first part contains a detailed non-axiomatic introduction to set theory. This introduction is carried out on a quite precise, but intuitive level, initially presenting many of Cantor's original ideas, including those on defining cardinals and order types as abstract objects. Only later, in Sections 8-11, do we discuss von Neumann's definition of ordinals and prove results important even for mathematicians working in various areas other than set theory. This part is well suited for a one-semester undergraduate course, and it is generally used in Hungarian universities. As is customary in mathematics textbooks at Hungarian universities, each assertion announced in the text is accompanied by a complete and detailed proof.

1.4. Review by: Jakub Jasinski.
Mathematical Reviews MR1728582 (2000m:03001).

From the introduction: "The book was prepared on the basis of courses and lectures by András Hajnal for mathematics majors at Roland Eötvös University in Budapest, Hungary."

Part I presents a traditional introduction to fundamental concepts of non-axiomatic set theory starting with cardinality and followed by order types, well-ordered sets, and the development of von Neumann ordinal numbers (called the "good sets''). Having those, the authors present the transfinite recursion theorem and establish the equivalence of AC with the well-ordering theorem, Teichmüller-Tukey lemma, and other classic results. Later, cardinals |A| are defined as least ordinals equivalent to A; their arithmetic properties and cofinality are discussed. Presentation of the generalised continuum hypothesis and Silver's theorem concludes Part I. An Appendix to Part I shows how the non-axiomatic presentation of Part I can be converted to a more rigorous axiomatic approach. Here some familiarity with mathematical logic is required. Proofs of relative consistency using models and large cardinals are outlined.

Part II is devoted to selected topics in combinatorial set theory, for example Ramsey's theorem and partition calculus. Using Skolem functions the authors show that cardinals are strongly inaccessible iff they are first-order strongly indescribable. Mahlo, measurable, weakly compact and Ramsey cardinals are presented. ...

The choice of topics provides for a natural flow where new concepts follow from the previously presented theorems. Each section is equipped with a few problems; hints are provided for the more difficult ones.

1.5. Comment by: Mirna Dzamonja.
András Hajnal, life and work, News Bulletin of the Iranian Association for Logic.

My students in England and in France invariably get this book on their reading list and they love it.

2.1. From the Publisher.

This work presents the most important combinatorial ideas in partition calculus and discusses ordinary partition relations for cardinals without the assumption of the generalised 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.

2.2. From the Preface.

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 generalisations of 'this theorem. The first important generalisation was the Erdős-Dushnik-Miller theorem which says that for an arbitrary infinite cardinal $\kappa$, if we put the edges of a complete graph of cardinality $\kappa$ into two classes then either the first class contains a complete graph of cardinality $\kappa$ or the second one contains an infinite complete graph. An earlier result of Sierpinski says that in case $\kappa = 2^{\aleph _{0}}$ 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 Erdős and Rado in 1956. In 1965, Erdős, Hajnal, and Rado gave an almost complete discussion of the ordinary partition relation for cardinals under the assumption of the generalised continuum hypothesis. At that time there were hardly any general results for ordinals, though there were some results of Specker, and the paper of Erdős 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 Erdős 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 generalised 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.

2.3. Review by: Carlos A di Prisco.
Mathematical Reviews MR0795592 (87g:04002).

The combinatorial theory of infinite sets constitutes now by itself a branch of mathematics with its own methods and applications to other areas. It includes topics which have arisen as generalisations to the infinite of ideas developed originally for finite sets or structures (such as graphs and partial orders), as well as topics which are inherent to infinite sets.
...
The book under review, as its subtitle indicates, deals almost exclusively with partition relations and constitutes a comprehensive monograph on the subject. Its appearance is welcome and fills a long-standing gap in the contemporary set-theoretical literature.

The authors state in their preface that their aim in writing the book is to present the most important combinatorial ideas in the partition calculus and to give a discussion of the ordinary partition relation for cardinals without the assumption of the generalised continuum hypothesis. The aim is amply satisfied; the authors single out the main types of arguments, fully exploring their power to obtain an immense amount of results.

The axiom of choice is assumed throughout the book, and the authors choose not to bother with ways to avoid using it in cases in which this axiom is not necessary. There is only brief mention of infinite exponent partition relations.

The proofs given in the book are always elegant and nicely presented, although on a few occasions the desire to state results of great generality might cause some pain to the inexperienced reader.

The reader is assumed to be familiar with the basic ideas of set theory. There is an introductory chapter (Chapter 1) containing the main basic set-theoretical concepts and results used in the book. Chapter 2 is also devoted to some preliminaries, including stationary sets and equalities and inequalities for cardinals.

The partition symbols are introduced in Chapter 3. In addition to the ordinary partition properties, the square bracket partitions, the polarised partitions, and other types of partition properties are considered in full generality. Section 8 in this chapter is very useful as a reference; the authors do well in advising the reader to skip this section and use it only to look for definitions of partition symbols used elsewhere in the book. Ramsey's theorem is proved along with its finite version, as well as some other classical results.

Chapters 4 and 6 develop the main tools used to prove partition relations. Chapter 4 contains a general framework for tree arguments and the proof of the "stepping up lemma''. Chapter 6 deals with "canonisation'' of partitions. Chapter 5 is a vast collection of negative results, including partition relations for finite sets.

Chapter 7 establishes some connections between large cardinals and partition relations. ...

Chapters 8 and 9 are the most technical in character. They contain the collection of "all the results known to the authors'' (i.e. all the results) for ordinary partition relations, systematically organised. The authors announce that "one might expect little enjoyment from reading these chapters'', a wise warning for the leisurely reader. The importance of these two chapters, the authors explain, lies in providing practical tests to decide whether a partition relation holds and also in locating the most important open problems.

Chapter 10 is devoted to applications of the combinatorial methods to topology, set mappings, cardinal exponentiation and saturated ideals. It is a very enjoyable chapter, a nice reward after the hardships of the previous ones.

The book ends with a survey of results on square bracket partition relations.

2.4. Review by: Neil H Williams.
The Journal of Symbolic Logic 53 (1) (1988), 310-312.

Modern combinatorial set theory could be loosely (and somewhat inaccurately) described as the study of partition relations; the various partition symbols in their general form provide a succinct formulation of a wide variety of combinatorial problems.
...
The setting for the book is classical Zermelo-Fraenkel set theory with the axiom of choice (ZFC). There are two, rather different, ways in which progress on a partition relation may be blocked in ZFC. The first is that some of the partition properties are large cardinal properties so the existence of uncountable cardinals with these properties is unprovable in ZFC. The second is that the required result may be consistent with, and independent of, the ZFC axioms. The book covers details of the appropriate large cardinal properties, but not of the independence results. Thus there are no proofs obtained by forcing, or deductions from extra axioms such as Martin's axiom or $\Diamond$. Mention is frequently made in passing of consistency results, to indicate how close the results presented come to being the best possible, but the reader will need to look elsewhere for the details of the proofs. The book opens with two chapters that review the standard results in set theory on which the book is based: ordinals, cardinals, some equivalents of the axiom of choice, and results concerning cardinal exponentiation. Chapter 3 begins with a guide to partition symbols, giving the definition of the more important symbols that have appeared in the literature, though many of these do not appear subsequently in the book. The rest of the chapter is concerned with the basic properties of the ordinary partition symbol ...
...
By the nature of the task the authors have given themselves, the book is not entirely easy reading. Some results, of necessity, are stated and proved in considerable generality. The authors have attempted to ease the reader's path by including some illuminating comments on what they aim to do, but the going can still be tough.