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A remark on the homogeneity of infinite permutation groups. (English) Zbl 0733.03040

A permutation group G acting on a cardinal \(\kappa\) is called \(\lambda\)- homogeneous, where \(\lambda\) is a cardinal \(<\kappa\), if for all \(A,B\in [\kappa]^{\lambda}\) there is a \(g\in G\) such that \(g[A]=B\). This paper is about the phenomenon that \(\lambda\)-homogeneity can fail to imply \(\mu\)-homogeneity for \(\mu <\lambda\). While it has been shown by P. M. Neumann [Bull. Lond. Math. Soc. 20, 305-312 (1988; Zbl 0644.20007)] that if \(\mu <\omega \leq \lambda <\kappa\), then any permutation group acting on \(\kappa\) that is \(\lambda\)-homogeneous is \(\mu\)-homogeneous as well, examples that are \(\omega_ 1\)-homogeneous but not \(\omega\)- homogeneous have been constructed under \(MA+``2^{\omega}\) is large” by P. Nyikos and independently by S. Shelah and S. Thomas. The author sheds some more light on the phenomenon by proving that if \(\kappa >\omega\) and the combinatorial principle \(\square_{\kappa}\) holds, then there is a \(\kappa\)-homogeneous permutation group acting on \(\kappa^+\) that is not \(\lambda\)-homogeneous for any \(\lambda\) with \(\omega \leq \lambda <\kappa\). Thus homogeneity is not necessarily preserved downwards even if CH holds. One question that the author’s result still leaves unanswered is that of whether or not the existence of an \(\omega_ 1\)-homogeneous permutation group acting on \(\kappa \geq \omega_ 3\) that is not \(\omega\)-homogeneous is consistent with CH.
Reviewer: J.Takahashi (Kobe)

MSC:

03E35 Consistency and independence results
20B07 General theory for infinite permutation groups
03E05 Other combinatorial set theory

Citations:

Zbl 0644.20007
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