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Claim. Theorem 2 of Jafar Shaffaf, A solution of the Erdős-Ulam problem on rational distance sets assuming the Bombieri-Lang conjecture, Discrete Comput. Geom. 60 (2018), no. 2, 283-293, states that, assuming the Bombieri-Lang conjecture, there is no dense rational distance subset of the plane, which would answer Problem 212 no. The tool is Theorem 1: for an even number $m=2g+2\ge 6$ of points $(\alpha_i,\beta_i)$ in the plane, not all on a line, the distance surface

$$z^2=\prod_{i=1}^{m}\bigl((x-\alpha_i)^2+(y-\beta_i)^2\bigr),$$

projectivized in $\mathbb{P}^3$, is a surface of general type. For Theorem 2 the paper takes a dense rational distance set $S$ containing $(0,0)$ and $(1,0)$, whose points then have the form $(a,b\sqrt k)$ with $a,b$ rational and $k$ a squarefree integer (Lemma 2), chooses six points of $S$ in general position, and considers their distance surface $X$ over $K=\mathbb{Q}(\sqrt k)$. By Theorem 1, $X$ is of general type, so the Bombieri-Lang conjecture forbids a Zariski-dense set of $K$-rational points on it; but $S$ is Zariski dense in the plane and $X$ is birationally a double cover of $\mathbb{P}^1\times\mathbb{P}^1$, so $S$ lifts to a Zariski-dense set of $K$-points of $X$, a contradiction. Corollary 1 draws from the same argument, with the Solymosi-de Zeeuw theorem that lines and circles are the only irreducible plane algebraic curves carrying an infinite rational distance set, that a rational distance set with infinitely many points not all on a line has all but at most four of its points on a line or all but at most three on a circle. The library card shaffaf_2018_solution_erdos_ulam_problem_rational_distance records Theorems 1 and 2 and Corollary 1. Terence Tao's blog post of 20 December 2014 derives the same conditional answer independently (its page); the post mentions an unpublished result of Shaffaf, and the preprint of this paper was posted to arXiv eleven days later.

Hypothesis. The paper states the hypothesis as the Weak Lang conjecture: for a projective variety of general type defined over a number field $K$, the set of $K$-rational points is not Zariski dense. Theorem 2 uses it for one surface over the field $\mathbb{Q}(\sqrt k)$. The conjecture is unproven, so the claim is conditional and derives no standing for the problem by itself; the paper presents the result as a witness to the strength of the Bombieri-Lang conjecture rather than a proof of the Erdős-Ulam conjecture.

Acceptance. Refereed: Discrete & Computational Geometry, volume 60, issue 2 (September 2018), pp. 283-293, published online 4 May 2018; the preprint is arXiv:1501.00159, first posted 31 December 2014. The site labels the problem OPEN, so the curator's remark that Shaffaf obtained the conditional answer is commentary on an open problem and not acceptance, and no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.