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Book Reviews Pe~iodica Mochem¢,~,a Hunga~iea Vol. 18 (1), (1987), pp. 81---85 BOOK REVIEWS P. Erd6s, A. Hajnai, A. M~t~ and R. Rado, Combinatorial set theory: Partition relations for cardinals (I)isquisitiones Mathematicae TTungaricae, 13), 347 pages, Akad6- mlai Kiad6, Budapest, 1984 [(Studies in Logic and the Foundation of Mathematics, 106), North.Holland, Amsterdam, 1984]. Like true love, combinatorial set theory is easy to recognize but impossibly hard to define. In his book Combinatoricd set flurry [2], Neff Williams says, "Combinatorial theory is largely the study of properties that a set or fArn!!y may have by virtue of its cardinality, although this may be ~wdened" to consider related properties,, held by sets carrying a simple structure, such as ordered or well-ordered sets . When applied to infinite sets this definition is as good as any other, but it is at once too wide and too narrow. It is too wide because other areas of set theory, like axiomatic or descriptive set theory, also deal with properties that sets may have by virtue of their cardinality, and it is too narrow because combinatorial set theory often deals with quite complicated structures on sets as well as with simple ones. Combinatorial set theory is sometimes referred to as Hungarian set theory in view of the central role played in its development by ErdSs, Hajnal and many other Hungarians, and it is sometimes called infinitary combinatorics, usually in the hope that if one knows what finite eombinatorics is then the nature of the infinite version should be clear. In fact combinatorial set theory has rather little in common with finite combina- torics. Some idea of its extent can be gained by perusing the contents of [2], which is intended as an introduction to the "classical" part of combinatorial set theory. There are sections on almost disjoint families, transversals (i.e., one-to-one choice functions), ordinary partition relations, set mappings, polarized partition relations, the theory of infinite graphs (especially problems about chromatic number), decomposition and inter- section properties of sets, and partition relations for ordinal numbers A more modern treatment would include results about trees of transfinite height as well as propositions consistent with and independent of the usual axioms for set theory; many feel that much of the theory of forcing and generic sets is combinatorial in nature. But the heart of hearts of combinatorial set theory is the partition calculus, first formulated explicitly by ErdSs and Rado in their famous 1956 paper [1 ]. It is the parti- tion calculus that gives combinatorial set theory not only its unique content but also its i~niqU~tetnY~iOg~arPdhisCnalo~Ptil~oarance.The "ordinary" partition relation for cardinal numbers and has the following meaning. IfA is ~ set and n is a cardinal number let [,4]n denote the set of n-element subsets of ,4. Then ~ -~ (~t)~ means that ~, ~t, p and n are cardinal numbers, and for any function f: [~]n ..~/~ (f is to be regarded as a pa~ion of the n-ele- ment subsets of x) there is A c_ ~ such that A has cardinaHty Z and f is constant on L4] n. The first result in the partition calculus is Rarnsey's Theorem, which may be expressed as s 0 --~ (s0)~ for all finite n and/¢. 6 Periodica Math. 18 (1) AkacTJm~aiKiadd, B~pest D. Reidel, Dordreeht 82 BOOK REVIEWS In their book Combinatorial ~et theory : Partition relations ]or cardinals, ErdSs, I-Iajnal, M~t4 and Rado begin with Ramsey's Theorem and continue through a full treatment of ordinary partition relations. For reasons which have never been completely clear, this subject is remarkably free of the concern with consistency and independence results that clutter many other parts of combinatorial set theory. A pleasant by-product is the fact that the book is really quite complete. The authors begin with fundamentals like Ramsey's Theorem and the fact that the partition relation ~ ~ (~t)~ is never true for infinite n (assuming the Axiom of Choice), and proceed immediately to the tree-theoretic techniques for establishing positive parti- tion relations. A long chapter is devoted to the counterexamples necessary to demonstrate the sharpness of the positive results, the case of partition relations for singular cardinals is treated, and then large cardinals are briefly attacked. Unfortunately, only weakly compact cardinals and measurable cardinals are studied; this is in my opinion the only real lack in the book. Weakly compact cardinals are those ~ satisfying ~ -~ (~)~ (equiv- alently ~ -~ (~)~ for all finite n and all ~t ~ ~). Measurable cardinals ~ have a more powerful property expressed as ~ -~ (~)<% i.e.,~.a_secluence of functions ],: [~]" --~ 2 is specified then there is a single set A _ ~ of cardi~lity ~ sich that each ]n is constant on [A] n. A cardinal satisfying ~ --~ (~}':~ ~rcalled a Ramsey cardinal; measurable cardinals are Ramsey but not every Ramsey cardinal is measurable. Ramsey cardinals are a sub- class of the wider class of E,rd6s cardinals, and it is this class that is responsible for some of the most spectacular applications of the partition calculus outside combinatorial set theory, namely the work of Silver, Solovay and others on indiscernibles over the con- structible universe and the real number 0~. It is to be regretted that there is still no easily accessible treatment of ErdSs cardinals in print. Two full chapters of the book are devoted to collating and summarizing the results on ordinary partition relations. To the best of my knowledge this sort of thing has never appeared before. It should be immensely useful to student and expert alike. The book concludes with some applications of the partition calculus, including results on cardinal exponentiation at singular cardinals, together with a very brief treat- ment of the so-called square bracket relation ~ -~ [A]~, which is defined like ~ ~ (A). except that instead of requiring f to be constant on [A] a one requires that / omit at lcas~ one value on [A]n. Several of these results are now obsolete in view of TodorSevid's recent result (in ZFC) that ~1 ~ Isling, but the methods remain of interest. The book is generally quite well-written; its tone is informal and leisurely. It should remain the standard reference for ordinary partition relations for a long time. REFERENCES [1] P. ERD6S and R. l~xoo, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427--489. MR 18, 458 [2] N. H. WIT.LIAMS, Combinatorial set theory, North-Holland, Amsterdam, 1977. Zbl 362:04008 J. E. Baumgartner (Hanover, N. H.) Rings of continuous functions (Lecture Notes in Pure and Applied Mathematics, 95), edited by CH.E. A~, x -~ 318 pages, M. Dekker, New York, 1985. This book is based on the material of a special session on Rings of Continuous Functions held at the Annual Meeting of the American Mathematical Society in Cin- cinnati, Ohio. More than twenty years after the classical monograph by L. Gillman and M, Jerison with the same title, the reader of the present book gets an excellent survey both on the history and on the recent development of the subject. Special emphasis is given to categorical methods and to problems connected with deep results in set theory. in particular with cardinal functions. The papers contained in the book are due to CH. E AvLL, R. L. BLAIR, D. E. CAMERON, W. WISTA_RCOMFORT and T. RETTA, A. Dew, W. A° FELDMAI~, J. F. PORTER, R. FOX, L. GrLLMA~, A. W. HAGER, M. E. HENRIKSE~, M. .
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