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Set mappings and polarized partition relations. (English) Zbl 0215.32903

Combinat. Theory Appl., Colloq. Math. Soc. János Bolyai 4, 327-363 (1970).
From the introduction: [For the entire collection see Zbl 0205.00201.]
A set mapping on a set \(S\) is a function \(f\) from \(S\) into the set of subsets of \(S\) such that \(x \not\in f(x)\) \((x \in S)\); \(A \subset S\) is called a free set (for the set mapping) if \(y \not\in f(x)\) for all \(x,y \in A\), i.e. \(A \cap f(A) = \emptyset\).
In this paper we shall consider set mappings on a well-ordered set \(S\) in the case when the order type of \(S\) is not necessarily an initial ordinal. In particular, we examine the truth status of the following statement \(SM(\alpha , \lambda)\). If \(f\) is any set mapping of order \(\alpha\) on a set type \(\lambda\), then there is a free subset having the same order type \(\lambda\). The Erdős-Specker generalization of the Ruziewicz conjecture asserts that \(SM(\alpha,\lambda)\) holds if \(\lambda\) is an infinite initial ordinal and \(\alpha < \lambda\). We only examine the problem for the case when \(|\lambda| = \aleph_1\) although some of our results hold more generally. We will prove that \(SM(\alpha,\lambda)\) holds in the following cases:
(i) \(\alpha < \omega_1\) and \(\lambda = \omega^{\sigma_1+1}_1+ \ldots +\omega^{\sigma_k+1}_1 < \omega^{\omega+2}_1\) (\(k\) finite);
(ii) \(\alpha = \omega_0\) and \(\lambda = \omega_1\gamma < \omega^{\omega+2}_1\);
(iii) \(\alpha < \omega_0\); \(\lambda = \omega\Theta\), where \(\Theta\) is arbitrary.
Note that the form given for \(\lambda\) in (i) is the most general for which \(SM(\alpha,\lambda)\) is true with any \(\alpha < \omega_1\).

MSC:

03E05 Other combinatorial set theory

Citations:

Zbl 0205.00201