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9.41 (1984)

Partially Solved

Let $\Omega$ be a countably infinite set. For $k \geqslant 2$ define a $k$-section of $\Omega$ to be a partition of $\Omega$ as union of $k$ infinite sets.
$\qquad$ a) Does there exist a transitive permutation group on $\Omega$ that is transitive on $k$-sections but intransitive on ordered $k$-sections?
$\qquad$ b) Does there exist a group that is transitive on $k$-sections but not on $(k + 1)$-sections?
$\qquad$ c) Does there exist a transitive permutation group on $\Omega$ that is transitive on $\aleph_0$-sections but which is a proper subgroup of $\operatorname{Sym}(\Omega)$?

Contributor: P. M. Neumann

Progress

a) Yes, there does. Let $U$ be a non-principal ultrafilter in $\mathcal{P}(\Omega)$ and let $G = \{g \in \text{Sym}(\Omega) \mid \text{Fix}(g) \in U\}$. It is not hard to prove that $G$ is transitive on $k$-sections but not on ordered $k$-sections for any $k$ in the range $2 \leqslant k \leqslant \aleph_0$. (P. M. Neumann, Letter of October, 5, 1989.)
c) Editors’ comment: An affirmative answer is consistent with the ZFC axioms of set theory (S. M. Corson, S. Shelah, Preprint, 2025, https://arxiv.org/abs/2503.12997).

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