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Updated Oct 8

Problem 212

claims/: The 2 claim pages of Problem 212, one per claimant's result; the problem's standing derives from them.

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

Status. 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.

Source. erdosproblems.com/212, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #212, https://www.erdosproblems.com/212.

References.

[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.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (19)

Tao's conditional answer under the Bombieri-Lang conjecture Shaffaf's conditional answer under the Bombieri-Lang conjecture problems/distance_problems/E0212/claims Distance Problems discrete_geometry/erdos_1987_combinatorial_metric_problems_geometry Ulam's conjecture, p. 172: an infinite plane set with all distances rational cannot be everywhere dense Distance Problems distance_problems/anning_1945_integral_distances Construction (pp. 598-599): a set dense on a circle with all distances rational, and Ulam's question distance_problems/ascher_2019_erdos_ulam_problem_lang_s_conjecture Proposition 3.6 (p. 5): under Lang's Conjecture, rational distance sets lie in proper algebraic sets of bounded total degree Proposition 5.1 (p. 8): the distance surface cut out by m quadrics is of general type for m at least 4 distance_problems/shaffaf_2018_solution_erdos_ulam_problem_rational_distance Corollary 1 (p. 3): an infinite rational set not on a line has all but 4 points on a line or all but 3 on a circle Theorem 1 (p. 3): the distance surface of 2g + 2 points not all on a line is of general type Theorem 2 (p. 3): under the Bombieri-Lang conjecture no dense subset of the plane has all distances rational distance_problems/solymosi_2010_question_erdos_ulam Theorem 2.1 (p. 2): a rational set meets a real algebraic curve finitely often unless the curve has a line or circle component Theorem 2.2 (p. 2): a rational set with infinitely many points on a line or circle has at most 4 or 3 points off it

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