19.89 (2018)

A permutation group is subdegree-finite if every orbit of every point stabiliser is finite. Let $\Omega$ be countably infinite, and let $G$ be a closed and subdegree-finite subgroup of $\text{Sym}(\Omega)$ regarded as a topological group under the topology of pointwise convergence.

Conjecture: If the minimal degree of $G$ is infinite, then there is some subset of $\Omega$ whose setwise stabiliser in $G$ is trivial.

Note that this conjecture implies Tom Tucker’s well-known Infinite Motion Conjecture for graphs.