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Claim. Theorem 2 of P. Erdős and S. Kakutani, On non-denumerable graphs, Bull. Amer. Math. Soc. 49 (1943), no. 6, 457--461, says that the continuum hypothesis holds if and only if $\mathbb{R}$ is a union of countably many sets each linearly independent over $\mathbb{Q}$ (the corpus's source card). In such a set no distance repeats, because two distinct pairs at the same distance would give a nontrivial rational relation among at most four of its points. So in every model of CH the answer to Problem 1127 for $n=1$ is yes, and ZFC does not refute it.

Covers. The case $n=1$ under CH. The theorem's converse half concerns rationally independent sets and does not show that CH is needed for distinct distances. That is Davies's Theorem 2, on Davies's page, which settles $n=1$ and $n=2$ completely.

Depends on. No page of this wiki.

Acceptance. Refereed: the result is a journal paper in the Bulletin of the American Mathematical Society. Reviewed: the curator of erdosproblems.com, T. F. Bloom, labels Problem 1127 independent and credits Erdős and Kakutani with the case $n=1$ under CH (problem page last edited 30 December 2025). The curator is independent of the authors.