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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 204.00905

**Autor: ** Erdös, Paul; Rado, R.

**Title: ** Partition relations and transitivity domains of binary relations (In English)

**Source: ** J. Lond. Math. Soc. 42, 624-633 (1967).

**Review: ** The main theorem is Theorem 2: For all positive integers m and n, for some positive integer l(m,n), for each ordinal number \alpha, \omega_{\alpha} l(m,n) ––> (m,\omega_{\alpha} n)^{2}; if l_{\alpha}(m,n) is the least such l(m,n) for a given \alpha, then \gamma \mapsto(m,\omega_{\alpha} n)^{2} for each \gamma > \omega_{\alpha} l_{\alpha} (m,n), and l_{\alpha}(m,n) \leq (2n-3)^{-1}[2^{m-1}(n-1)^{m}+n-2]; if m > 1, then l_{0}(m,n) \leq l_{\alpha} (m,n).

This generalizes some of Theorem 1, which handles the case \alpha = 0 and was proved by the same authors [Bull. Am. Math. Soc. 62, 427-489 (1956; Zbl 071.05105), Theorem 25]; Theorem 1 includes also a characterization of l_{0}(m,n). Among other results, Theorem 4 is an extension (the statement of which is not the obvious one) to infinite a of a result attributed to R. Stearns: If a is a finite cardinal, if \prec is a relation trichotomous on a set S, then \prec is transitive on some subset of S having cardinal a provided that |S| \geq 2^{n-1}.

**{**Reviewer's remarks:

(1) It is easily seen that l_{0}(m,2) as characterized by Theorem 1 is the least integer l such that for each set S with cardinal \geq l, for each relation \prec trichotomous on S, \prec is transitive on some subset of S having cardinal a. Specializing the estimate for l_{\alpha} (m,n) in Theorem 2 to n = 2 yields the Stearns result.

(2) Theorem 1 is slightly misstated. If m = 1, ``\gamma \mapsto (m,\omega_{0}n)^{2}'' should be changed by replacing \gamma by \omega_{0}(l_{0}(m,n)-1).

(3) In the footnote on p. 625 ``(n-1)^{\mu}'' should be replaced by ``(2(n-1))^{\mu}''.

(4) On p. 627, (i) is not quite adequate but becomes so on replacing ``for x in A'_{\beta}'', by ``for each x in A'_{\beta} and |U_{0}(x)A_{\gamma}| = \aleph_{\alpha} for some x in A_{\beta}'', and (ii) is not quite adequate but becomes so on inserting ``and |U_{0}(x)A_{\gamma}| < \aleph_{\alpha} for some x in A_{\beta}'' after ``|U_{0}(\bar x)A_{\gamma}| = \aleph_{\alpha}''. [The iteration of the operators O_{\lambda} then becomes adequate for the task at hand.] The last sentence of the third paragraph of (ii) appears to be an inaccurate oversimplication. [It is clear from the rather involved proof of Theorem 2, to which these points attach, that these inaccuracies were not in the original thinking things through. There are only a few others, which are more easily spotted.]

(5) In line with (1) above and the discussion of l_{0}(m,n) on p. 624, the condition under (i) in Theorem 4 is best possible not only for 1 \leq a \leq 3 but also for 1 \leq a \leq 4**}**.

**Reviewer: ** A.H.Kruse

**Classif.: ** * 05D10 Ramsey theory

03E05 Combinatorial set theory (logic)

04A20 Combinatorial set theory

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