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Source. Problem 25, p. 12 (Section 6, "Subsets with no repeated distances", pp. 10--13), with Table 2 (p. 11), 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. 12). $\mathsf{subset}_d(n)$ is the largest number such that every set of $n$ points in $\mathbb R^d$ contains a subset of $\mathsf{subset}_d(n)$ points spanning no distance more than once; it is the higher-dimensional form of $\mathsf{subset}(n)$ from Problem 22.

What the survey records (p. 12), none of it proved in the survey:

• Lower bounds: $\mathsf{subset}_d(n)=\Omega(n^{1/(3d-2)})$ (Thiele's thesis, Theorem 4.33), improved by Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky to $\mathsf{subset}_d(n)=\Omega\bigl(n^{1/(3d-3)}(\log n)^{1/3-2/(3d-3)}\bigr)$.
• Upper bound: $\mathsf{subset}_d(n)\le\mathsf{subset}(\mathcal L_d)=O(n^{1/d})$, where $\mathcal L_d$ is an $n^{1/d}\times\cdots\times n^{1/d}$ integer lattice, which spans $O(n^{2/d})$ distinct distances.

Problem 25 (p. 12). "Find the asymptotic value of $\mathsf{subset}_d(n)$ for $d\geq3$." (quoted)

Read depth

Claims checked on the print. The cited bounds are reported as the survey states them and were not checked against their sources here.

Bears on

• Problem 1208: for $d\ge3$ the problem's $F_d(n)$, read as the largest size such that every $n$ points of $\mathbb R^d$ contain that many points with all distances distinct, is $\mathsf{subset}_d(n)$, and the problem asks for its estimate for fixed $d$. The survey records the bounds above and leaves the asymptotic value open as Problem 25.