Problem 1127

Can $\mathbb{R}^n$ be decomposed into countably many sets, such that within each set all the pairwise distances are distinct?

Status

SolvedIndependent

An accepted solution exists. The statement can be neither proved nor disproved from the standard axioms of set theory.

Da72: Davies, Roy O., Partitioning the plane into denumerably many sets without repeated distances. Proc. Cambridge Philos. Soc. (1972), 179-183.
Er81b: Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book).
ErKa43: Erdős, P. and Kakutani, S., On non-denumerable graphs. Bull. Amer. Math. Soc. (1943), 457-461.
Ku87: Kunen, Kenneth, Partitioning Euclidean space. Math. Proc. Cambridge Philos. Soc. (1987), 379-383.