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9.42 (1984)

Open

Let $\Omega$ be a countable set and let $D$ be the set of total order relations on $\Omega$ for which $\Omega$ is order-isomorphic with $\mathbb{Q}$. Does there exist a transitive proper subgroup $G$ of $\operatorname{Sym}(\Omega)$ which is transitive on $D$?

Contributor: P. M. Neumann

Progress

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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