Problem 213

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claims/: The 3 claim pages of Problem 213, one per claimant's result; the problem's standing derives from them.

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Statement. Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?

Formulation. The statement fixes $n\ge4$ and asks for $n$ points. The page reads it as asking whether such a set exists for every $n\ge4$, the reading of Erdős's 1983 lecture (Math. Chronicle 12 (1983), 35–54, p. 43, carded as erdos_1983_combinatorial_problems_geometry), which asks for $n$ points in general position and records that the case $n=5$ was settled and the general case, even $n=6$, was open, and the reading of the formal-conjectures statement, which quantifies over all $n\ge4$. Under this reading a construction answers single instances yes, and a uniform bound on the size of such sets would answer no. The property passes to subsets: a subset of a set with no three points on a line, no four on a circle and all distances integers keeps all three properties, so a construction for $n$ points answers every smaller $n\ge4$ as well.

Status. Open, the site's label (OPEN). Three claim pages are recorded, two accepted partial claims and one accepted conditional claim, none of which settles the question. [[problems/distance_problems/E0213/claims/1971_01_01_harborth|Harborth's five points]] and [[problems/distance_problems/E0213/claims/2007_09_29_kreisel_kurz|Kreisel and Kurz's seven points]] have the three properties, so the answer is yes for $n=5$ and for every $n\le7$, and no construction with eight points is known. [[problems/distance_problems/E0213/claims/2019_01_09_ascher_braune_turchet|Ascher, Braune and Turchet]] prove that Lang's conjecture, which is unproven, implies a uniform bound on the size of such sets, so that under it the answer would be no for all large $n$. Two further results settle no instance and are recorded here rather than as claims: Anning and Erdős [AnEr45] proved that an infinite set of points in the plane with all distances integers lies on a line, and Greenfeld, Iliopoulou and Peluse [GIP24] proved unconditionally, in their Corollary 1.3, that a set with the three properties inside $[-N,N]^2$ has $O((\log N)^{O(1)})$ points. The site's remarks also point to Problem 130.

Source. erdosproblems.com/213, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #213, https://www.erdosproblems.com/213.

References.

• [ABT20] Ascher, K. and Braune, L. and Turchet, A., The Erdős-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).
• [AnEr45] Anning, Norman H. and Erdős, Paul, Integral distances. Bull. Amer. Math. Soc. (1945), 598-600.
• [GIP24] Greenfeld, R. and Iliopoulou, M. and Peluse, S., On integer distance sets. arXiv:2401.10821 (2024).
• [KK08] Kreisel, Tobias and Kurz, Sascha, There are integral heptagons, no three points on a line, no four on a circle. Discrete Comput. Geom. 39 (2008), 786-790.

Formalization. Statement in formal-conjectures.

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.

• anning_1945_integral_distances
• anning_1945_integral_distances / theorem_p598
• ascher_2019_erdos_ulam_problem_lang_s_conjecture
• ascher_2019_erdos_ulam_problem_lang_s_conjecture / corollary_1_2
• ascher_2019_erdos_ulam_problem_lang_s_conjecture / definition_p1
• ascher_2019_erdos_ulam_problem_lang_s_conjecture / proposition_5_1
• ascher_2019_erdos_ulam_problem_lang_s_conjecture / theorem_1_1
• erdos_1983_combinatorial_problems_geometry
• erdos_1983_combinatorial_problems_geometry / problem_p43
• erdos_1983_combinatorial_problems_geometry / theorem_p42
• greenfeld_2024_integer_distance_sets
• greenfeld_2024_integer_distance_sets / corollary_1_3
• greenfeld_2024_integer_distance_sets / theorem_1_1
• kreisel_2008_there_are_integral_heptagons_no_three
• kreisel_2008_there_are_integral_heptagons_no_three / example_p3
• kreisel_2008_there_are_integral_heptagons_no_three / question_p4
• kreisel_2008_there_are_integral_heptagons_no_three / theorem_1