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**Zentralblatt MATH**

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

**Zbl.No: ** 271.04003

**Autor: ** Erdös, Paul; Milner, E.C.

**Title: ** A theorem in the partition calculus. (In English)

**Source: ** Can. Math. Bull. 15, 501-505 (1972); corrigendum ibid. 17, 305 (1974).

**Review: ** It is shown that if h < \omega and \nu < \omega_{1} then \omega ^{1+\nu h} ––> (2^{h}, \omega ^{1+\nu})^{2}, as known to the authors since 1959. The literature of this result and related results are discussed. The proof is by induction on h from a more comprehensive theorem. For each ordered set S, tp \, S is the order type of S. Consider any order type \alpha. \alpha is right-AI if and only if \alpha = \beta+\gamma, \gamma \ne 0 implies that \gamma \geq \alpha (AI abbreviates additively indecomposable). \alpha is strongly indecomposable (briefly, SI) if and only if whenever A = B \cup C and \alpha is the type of A, B, or C has type \geq \alpha. For each order type \beta, \beta is strong if and only if whenever tp \, B = \beta and D \subset B there are n < \omega and subsets D_{1}, ... ,D_{n} of D such that tp \, D_{i} is SI for i = 1, ... ,n and such that for each M \subset D, if tp(M \cap D_{i}) \not \geq tp \, D_{i} for i = 1, ... ,n then tp \, M \approx tp \, D. For example, each ordinal number and its reverse are strong. The theorem proved is that for each SI and right-AI order type \alpha and each strong denumerable order type \beta, if 2 le k < \omega and \alpha ––> (k, \gamma)^{2} then \alpha \beta ––> (2k, \gamma \cup \omega \beta)^{2}. (It is added in proof that *F. Galvin* has proved the same thing with ``strong'' deleted from the hypothesis on \beta, thus settling a conjecture of the writers.)

**Reviewer: ** A.H.Kruse

**Classif.: ** * 04A10 Ordinal and cardinal numbers; generalizations

05A17 Partitions of integres (combinatorics)

05A99 Classical combinatorial problems

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