Claims

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1971_01_01_harborth: Harborth (Elem. Math. 1971) gives five points in the plane, no three on a line and no four on a circle, with all ten distances integers, answering the instances n = 4 and n = 5 yes.

2007_09_29_kreisel_kurz: Kreisel and Kurz (Discrete Comput. Geom. 2008) find by exhaustive search seven points in the plane, no three on a line and no four on a circle, with all distances integers and minimum diameter 22270; yes for every n up to 7.

2019_01_09_ascher_braune_turchet: Ascher, Braune and Turchet (Bull. London Math. Soc. 2020) prove that Lang's conjecture implies a uniform bound on rational distance sets in general position, so under that unproven hypothesis the answer is no for large n.

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