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Source. The unnumbered definition on p. 1 of Kenneth Ascher, Lucas Braune and Amos Turchet, The Erdős-Ulam problem, Lang's conjecture, and uniformity, arXiv:1901.02616v2 (17 August 2020), the version named on the source card.

Read depth. Claims checked: the definition, the seven-point example after it and the comparison with Kreisel and Kurz on p. 2 were read clause by clause on the printed pages. Nothing here is independently reviewed.

Statement

Definition (p. 1, unnumbered). A rational distance set is a subset of $\mathbb R^2$ in which the distance between any two points is rational. A subset $S\subseteq\mathbb R^2$ of cardinality $n$ is in general position when no subset of $S$ of cardinality $n-4$ lies on a line and no subset of $S$ of cardinality $n-3$ lies on a circle.

The paper motivates the definition by the result of Solymosi and de Zeeuw that a line (resp. circle) containing infinitely many points of a rational distance set contains all but at most four (resp. three) of its points (p. 1). It gives the example $n=7$: a seven-point set is in general position exactly when no three of its points lie on a line and no four on a circle (p. 1). On p. 2 it notes that for sets of more than seven points this notion is strictly weaker than the one used by Kreisel and Kurz, no three points on a line and no four on a circle.

Small sets (an observation of this page, not of the paper). Any set of at most two points lies on a line, so read literally the definition fails for $4\le n\le6$. For $n\ge7$ a set with no three points on a line and no four on a circle is in general position, since $n-4\ge3$ and $n-3\ge4$.

Proof pointer

A definition; nothing to prove.

Dependencies

None.

Bears on

• Problem 213: the problem's sets of $n\ge7$ points, no three on a line, no four on a circle and integer distances, are rational distance sets in general position in this sense, which is how Theorem 1.1 reaches the problem.