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Source. Problem 26, p. 13 (Section 6, "Subsets with no repeated distances", pp. 10--13), with the definition of $\mathsf{subset}'(n)$ on p. 12 and 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). $\mathsf{subset}'(n)$ is the largest number such that every set of $n$ points in the plane contains a subset of $\mathsf{subset}'(n)$ points spanning no isosceles triangle.

Problem 26 (p. 13), credited to Brass, Moser and Pach. "Find the asymptotic value of $\mathsf{subset}'(n)$." (quoted)

What the survey records (p. 13 and Table 2, p. 11):

• Lower bound $\Omega(n^{0.4315})$, by adapting the proof of Theorem 6.1 (see the Problem 22 page) with the count of repeated-distance quadruples removed, so that only Pach and Tardos's $O(n^{2.137})$ bound on isosceles triangles is used. The text writes this bound as "$s'(n)=\Omega(n^{0.4315})$" (p. 13, quoted), with $s'$ for $\mathsf{subset}'$.
• Upper bound, Table 2: $O(\sqrt n/(\log n)^{1/4})$, justified on p. 13 by "$\mathsf{subset}'(n)\leq s(n)=O\left(\sqrt{n}/(\log n)^{1/4}\right)$" (quoted), with $s(n)$ for $\mathsf{subset}(n)$.

On the upper bound. By the definitions, a subset in which no distance repeats spans no isosceles triangle, so every set has $\mathsf{subset}'(\mathcal P)\ge\mathsf{subset}(\mathcal P)$ and $\mathsf{subset}'(n)\ge\mathsf{subset}(n)$. The printed inequality runs the other way, so the survey's argument does not establish the upper bound listed in Table 2. This page records the bound only as printed.

Read depth

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

Bears on

• Problem 1207: for $d=2$ the problem's $P_2(n)$, read as the largest size such that every $n$ planar points contain that many points spanning no isosceles triangle, is $\mathsf{subset}'(n)$, and the problem asks in particular whether $P_2(n)<n^{1-c}$ for some $c>0$. The survey records the lower bound $\Omega(n^{0.4315})$ and leaves the asymptotic value open as Problem 26. Its Table 2 upper bound $O(\sqrt n/(\log n)^{1/4})$ would answer the particular question, but the argument printed for it does not hold, as stated above.