Problem 212

Is there a dense subset of $\mathbb{R}^2$ such that all pairwise distances are rational?

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label (OPEN). Two conditional claim pages record that the Bombieri-Lang conjecture would answer the question no: Shaffaf's refereed paper and Tao's blog post. The hypothesis is unproven, so neither page settles the problem.

ABT20: Ascher, K. and Braune, L. and Turchet, A., The Erdős-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).
Er87b: Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
SdZ10: Solymosi, Jozsef and de Zeeuw, Frank, On a question of Erdős and Ulam. Discrete Comput. Geom. 43 (2010), 393-401.
Sh18: Shaffaf, Jafar, A solution of the Erdős-Ulam problem on rational distance sets assuming the Bombieri-Lang conjecture. Discrete Comput. Geom. 60 (2018), 283-293.

Problem 212

Status

Proof claims

Discussion