b

is equivalent to the following condition. Let |X_{p}| = _{p} for p \leq s_{p} are disjoint, [X_{0},...,X_{s}]^{r0,...rs} = **{**X: X \subseteq X_{0} \cup ··· \cup X_{s},_{p}| = _{p}**}** = _{\nu < \lambda} J_{\nu}_{r} \subseteq X_{r}, for r \leq s and a \nu_{0} < \lambda such that |Y_{r}| = _{r\nu0}

"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 \aleph_{0}, and if we assume the General Continuum Hypothesis, then our results are complete for r =

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_{\xi} < a'**}**_{\xi1} < a_{\xi2} for \xi_{1} < \xi_{2} < a', [S]^{r} = _{\nu < \lambda} J_{\nu}_{\sigma}, \sigma < a', |S_{\sigma}| = _{\sigma}_{\sigma} \subseteq S and for X,Y in **[**\bigcup_{\sigma < a'} S_{\sigma} **]**^{r}, the relations |X\cap S_{\sigma}| = _{\sigma}|_{0} < \lambda such that X,Y in J_{\nu0}.

Define \alpha \dot- 1 =

(R) \aleph_{\beta+(r-2)} ––> (b_{\xi})^{r}_{\xi} < \lambda,

(IA) b_{0} = _{\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} < \aleph_{0} 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 b_{1} \geq \aleph_{0}, 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) b_{0} > \aleph_{\alpha \dot- (r-2)}.

(IIB) b_{\xi} \leq \aleph_{\gamma}, \xi < \lambda, \alpha =

(IIC1) b_{0} = _{\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| = ^{r} = _{\xi < c} J_{\xi}_{\xi} are disjoint, then there are a set X \subseteq S and a number \xi_{0} < c (a set D \subseteq c) such that |X| = _{\xi0}^{r} \cap J_{\xi0} = ^{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}_{2r-1}.

(iii) If \aleph_{\alpha}' is \aleph_{0} or a measurable cardinal, then \aleph_{\alpha} ––> [\aleph_{\alpha}]^{r}_{c} for c > 2^{r-1} and \aleph_{\alpha} ––> [\aleph_{\alpha}]^{r}_{c2r-1} for c < \aleph_{\alpha}.

(iv) \aleph_{2} ––> [\aleph_{0}, \aleph_{1}, \aleph_{1} ]^{3}.

On the other hand, there are many open problems, e.g. \aleph_{2} ––> [\aleph_{1}]^{3}_{4}?, \aleph_{3} ––> [\aleph_{1}]^{2}_{\aleph2,\aleph0}?

In the second part, the authors investigate the polarized partition relation \binom{a}{b} ––> \pmatrix a_{0}, a_{1} \\ b_{0}, 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

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