is equivalent to the following condition. Let |X_p| = a_p for p \leq s, X_p are disjoint, [X₀,...,X_s]^r₀,...r_s = {X: X \subseteq X₀ \cup ··· \cup X_s, |X\cap X_p| = r_p for p \leq s} = \bigcup_{\nu < \lambda} J_\nu. Then there exist sets Y_r \subseteq X_r, for r \leq s and a \nu₀ < \lambda such that |Y_r| = b_r\nu0 for r \leq s and

"In this paper our first major aim is to discuss as completely as possible the relation I. Our most general results in this direction are stated in Theorems I and II, ... . If we disregard cases when among the given cardinals there occur inaccessible numbers greater than \aleph₀, and if we assume the General Continuum Hypothesis, then our results are complete for r = 2, ... . It seems that there are only two essentially different methods for obtaining positive partition formulae: those given in Lemma 1 and those given in Lemma 3 ... In Lemma 5 we state powerful new methods for constructing examples of particular partitions which yield negative I-relation. ... Our second major aim is an investigation of the polarized partition relation III."
The exact formulation of the Lemma 1 is complicated; its contents may be shortly formulated as follows: in every sufficiently great tree, in which from every edge goes out a small number of branches, there is a large branche.
The simplest canonization lemma (the Lemma 3 proved using the Generalized Continuum Hypothesis) may be stated as follows: Let |S| = a > a' (a' is the smallest cardinal with which a is cofinal); r \geq 1, a = \sup {a_\xi < a'}, a_\xi1 < a_\xi2 for \xi₁ < \xi₂ < a', [S]^r = \bigcup_{\nu < \lambda} J_\nu, \lambda < a. Then there are disjoint sets S_\sigma, \sigma < a', |S_\sigma| = a_\sigma, S_\sigma \subseteq S and for X,Y in [\bigcup_{\sigma < a'} S_\sigma ]^r, the relations |X\cap S_\sigma| = |Y\cap S_\sigma| for \sigma < a' are equivalent to the condition: there is a \nu₀ < \lambda such that X,Y in J_\nu0.
Define \alpha \dot- 1 = \alpha for \alpha limit \alpha \dot- 1 = \beta if and only if \alpha = \beta+1, cr(\alpha) = cf(cf(\alpha \dot- 1). Let us denote:
(R) \aleph_\beta+(r-2) ––> (b_\xi)^r_\xi < \lambda,
(IA) b₀ = \aleph_\beta,
(IB) b_\xi < \aleph_\beta for \xi < \lambda,
(CA) prod_{1 \leq \xi < \lambda} b_\xi \leq \aleph_cr(\beta),
(CB) prod_{\xi < \lambda} b_\xi < \aleph_\beta,
(D) r \geq 3, \beta > cf(\beta) > cf(\beta) \dot- 1 > cr\beta, b_\xi < \aleph₀ for 1 \leq \xi < \lambda.
The first main theorem may be stated as follows. Let \lambda \geq 2, 2 \leq r < b_\xi \leq \aleph_\beta for \xi < \lambda. Assuming the Generalized Continuum Hypothesis we have:
(i) If (IA) holds, (D) does not holds, then (R) implies (CA).
(ii) If (IA) holds and b₁ \geq \aleph₀, then (R) implies (CA).
(iii) If (IA) holds and \aleph_\beta' is not inaccessible, then (CA) implies (R).
(iv) If (IA) holds and b_\xi < \aleph_\beta ' for 0 < \xi < \lambda then (CA) implies (R).
(v) If (IB) holds, then (CB) is equivalent to (R).
Let us denote:
(IIA) b₀ > \aleph_{\alpha \dot- (r-2)}.
(IIB) b_\xi \leq \aleph_\gamma, \xi < \lambda, \alpha = \gamma+s, \gamma limit and s < r-2.
(IIC1) b₀ = \aleph_{\alpha \dot- (r-2)},
(IIC2) b_\xi < \aleph_{\alpha \dot- (r-2)} for \xi < \lambda.
(R0) \aleph_\alpha ––> (b_\xi)^r_{\xi < \lambda}.
The second main theorem: Let \lambda \geq 2, 2 \leq r < b_\xi \leq \aleph_\alpha for \xi < \lambda.
Assuming the Generalized Continuum Hypothesis we have:
(i) If (IIA) holds, then (R0) is false.
(ii) If (IIB) and (IIC1) hold, (R0) implies that \aleph_{\alpha \dot-(r-2)} is inaccessible.
(iii) If (IIB) and (IIC2) hold, then (R0) is equivalent to the condition prod_{\xi < \lambda} b_\xi < \aleph_{\alpha \dot-(r-2)}.
The proofs are based on Lemmas 1, 2, 3 and 5. The Lemma 2 and 5 are the stepping-up and stepping-down Lemmas respectively, i.e. they are of the form "if a ––> (b_\xi)^r_{\xi < \lambda}, then a^+ ––> (b_\xi+1)^r+1_{\xi < \lambda}" and "if a \not ––> (b_\xi)^r_{\xi < \lambda}, then 2^a \not ––> (b_\xi+1)^r+1_{\xi < \lambda}", respectively (of course, under some assumptions).
A great part of the paper is devoted to the study of relations IV and V. The relation IV: a ––> [b_\xi ]^r_{\xi < c} (relation V: a ––> [b]^r_c,d) is equivalent to the condition: whenever |S| = a, [S]^r = \bigcup_{\xi < c} J_\xi, where the J_\xi are disjoint, then there are a set X \subseteq S and a number \xi₀ < c (a set D \subseteq c) such that |X| = b_\xi0 and [X]^r \cap J_\xi0 = Ø (|X| = b, |D| \leq a and [X]^r \subseteq \bigcup_{\xi in D} J_\xi). Some results (assuming the Generalized Continuum Hypothesis):
(i) \aleph_\alpha+1 \not ––> [\aleph_\alpha+1 ]²_{\aleph_{\alpha+1}} for any \alpha.
(ii) Let r \geq 2 and \alpha > cf (\alpha). Then \aleph_\alpha \not ––> [\aleph_\alpha]^r_2^r-1.
(iii) If \aleph_\alpha' is \aleph₀ or a measurable cardinal, then \aleph_\alpha ––> [\aleph_\alpha]^r_c for c > 2^r-1 and \aleph_\alpha ––> [\aleph_\alpha]^r_c2^r-1 for c < \aleph_\alpha.
(iv) \aleph₂ ––> [\aleph₀, \aleph₁, \aleph₁ ]³.
On the other hand, there are many open problems, e.g. \aleph₂ ––> [\aleph₁]³₄?, \aleph₃ ––> [\aleph₁]²_{\aleph2,\aleph0}?
In the second part, the authors investigate the polarized partition relation \binom{a}{b} ––> \pmatrix a₀, a₁ \\ b₀, b1 \endpmatrix, i. e. a special case of the relation III. A complete discussion is given, however, the results are not complete. Many other relations and problems are studied, but it is impossible to give a full list of them here.
The paper is rather difficult to read and gives the impression of a condensed version of a monography.
Reviewer:  L.Bukovský
Classif.:  * 05D10 Ramsey theory
                   03E05 Combinatorial set theory (logic)
                   04A20 Combinatorial set theory
                   03-02 Research monographs (mathematical logic)
                   05E10 Tableaux, etc.
                   04A10 Ordinal and cardinal numbers; generalizations
Index Words:  set theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag

Books	Problems	Set Theory	Combinatorics	Extremal Probl/Ramsey Th.
Graph Theory	Add.Number Theory	Mult.Number Theory	Analysis	Geometry
Probabability	Personalia	About Paul Erdös	Publication Year	Home Page