Claims

../

1943_06_01_erdos_kakutani: Erdős and Kakutani (Bull. Amer. Math. Soc., 1943) prove that the continuum hypothesis splits the reals into countably many rationally independent sets, which have no repeated distances: the case n = 1 under CH.

1972_09_01_davies: Davies (Proc. Cambridge Philos. Soc., 1972) proves that the plane splits into countably many sets with all distances distinct if the continuum hypothesis holds, and that such a partition, even of the line, implies the hypothesis.

1987_11_01_kunen: Kunen (Math. Proc. Cambridge Philos. Soc., 1987) proves under the continuum hypothesis that real n-space, for every finite n, splits into countably many sets with all distances distinct; with Davies's converse, independent of ZFC.

---