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Partitioning Euclidean spaces by spheres in intermediate dimensions

Determine, for integers n ≥ 2 and m with n+2 ≤ m < 2n+1, whether the Euclidean space R^m can be partitioned into isomorphic copies of the n-sphere S^n, both with and without the Axiom of Choice, and ascertain whether such partitions exist even when allowing spheres of varying radii.

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Background

Topological and choice-based results show that R{n+2} can be partitioned into topological copies of Sn (in ZF) and R{2n+1} can be partitioned into isomorphic copies of Sn (in ZFC).

For the intermediate range of dimensions between these two bounds, the existence of such partitions is unresolved, regardless of whether one assumes the Axiom of Choice, and remains open even when the spheres are permitted to have varying radii.

References

To the best of the author's knowledge, the question of whether $Rm$ can be partitioned into isomorphic copies of $Sn$ is open for $n+2\leq m < 2n+1$ and $n\geq 2$, with or without the Axiom of Choice, and the same is open even when allowing different radii.

— Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals (2501.03131 - Fatalini, 6 Jan 2025) in Subsection “Overview of PUCs” (Subsection \ref{subsection: PUC lit review})