Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 1127

claims/: The 3 claim pages of Problem 1127, one per claimant's result; the problem's standing derives from them.

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

Status. Independent.

Source. erdosproblems.com/1127, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1127, https://www.erdosproblems.com/1127.

References.

[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.

Formalization. A Lean proof of the result is in Boris Alexeev's lean-proofs repository, with Codex and GPT-5.6 Sol named as its formal authors; [[problems/set_theory/E1127/claims/1987_11_01_kunen|the claim page]] links it at its pinned commit. The corpus has not built the file.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (10)

Erdős and Kakutani's rationally independent decomposition of the line Davies's partition of the plane without repeated distances Kunen's partition of Euclidean space under the continuum hypothesis problems/set_theory/E1127/claims Set Theory and Infinite Combinatorics discrete_geometry/erdos_1978_set_theoretic Theorem 2 (p. 127): if c > aleph_1, every countable cover of the line has a part with a repeated distance Set Theory and Infinite Combinatorics set_theory/erdos_1943_non_denumerable_graphs Theorem 2 (p. 459): CH holds if and only if the reals are a union of countably many rationally independent sets

Graph