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Source. Problem 8, p. 6 (Section 3, "Restricted point sets in $\mathbb R^2$", pp. 4--6), with Table 1 (p. 4), of Adam Sheffer, Distinct Distances: Open Problems and Current Bounds, arXiv:1406.1949v3 (2 July 2018), the edition read for the source card.

Statement

Notation (p. 6). A planar point set is in general position if no three of its points are collinear and no four are cocircular. $D_{\mathrm{gen}}(n)$ is the least number of distinct distances determined by $n$ points in general position.

What the survey records (p. 6):

• It is not known whether $D_{\mathrm{gen}}(n)=\Theta(n)$.
• Upper bound $D_{\mathrm{gen}}(n)=n2^{O(\sqrt{\log n})}$, credited to Erdős, Füredi, Pach and Ruzsa: lattice points of an integer grid in about $\sqrt{\log n}$ dimensions lying on a common hypersphere, projected to a generic plane.
• Lower bound $D_{\mathrm{gen}}(n)=\Omega(n)$, since general position has no three collinear points (Lemma 3.1).

Problem 8 (p. 6). "Find the asymptotic value of $D_{\mathrm{gen}}(n)$." (quoted)

Read depth

Claims checked on the print. The cited upper bound is reported as the survey states it and was not checked against its source here.

Bears on

• Problem 98: the problem's $h(n)$, read as the largest such bound, is $D_{\mathrm{gen}}(n)$ in the survey's notation, and the problem asks whether $h(n)/n\to\infty$. The survey records that it is not known whether $D_{\mathrm{gen}}(n)=\Theta(n)$, with bounds $\Omega(n)$ and $n2^{O(\sqrt{\log n})}$, and leaves the question open as Problem 8.