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Claim. Theorem 1.1 of K. Ascher, L. Braune and A. Turchet, The Erdős–Ulam problem, Lang's conjecture and uniformity, Bull. London Math. Soc. 52 (2020), no. 6, 1053–1063, states that if Lang's conjecture holds then there is a constant bounding the cardinality of every rational distance set in general position in the plane. A rational distance set is a set all of whose pairwise distances are rational, and the paper calls a set of $n$ points in general position when no $n-4$ of them lie on a line and no $n-3$ on a circle. A set of $n\ge7$ points with no three on a line and no four on a circle is in general position in this sense, and integer distances are rational, so under the hypothesis the sets that Problem 213 asks for exist only for $n$ below a fixed constant: the answer to the question, read as asking for every $n\ge4$, would be no. The proof lifts the points of such a set to rational points on curves and surfaces of general type, following Solymosi and de Zeeuw and Tao, and applies the uniformity theorems of Caporaso, Harris and Mazur for curves and of Hassett for surfaces. The source card ascher_2019_erdos_ulam_problem_lang_s_conjecture summarizes the argument.

Hypothesis. The claim is conditional on Lang's conjecture, the paper's Conjecture 2.2, which says that the rational points of a variety of general type over a number field are not Zariski dense; the site's remarks call it the Bombieri–Lang conjecture. It is unproven, so this page derives nothing for the problem's standing. The theorem gives no explicit value of the bound and decides no single instance of the question; the paper itself notes the seven-point example of [[problems/distance_problems/E0213/claims/2007_09_29_kreisel_kurz|Kreisel and Kurz]] and that no larger example is known.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Bulletin of the London Mathematical Society 52 (2020), no. 6, 1053–1063, published online 2020-06-29; the Crossref record of the DOI gives these data. The site's remarks record the conditional uniform bound with this paper as its source, but the site labels the problem OPEN, so that remark is commentary on an open problem and no reviewed evidence is listed.