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Source. Problem 28, p. 13 (Section 7, "Distinct distances with local properties", pp. 13--15), with Table 3 (p. 13), 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. 13). For positive integers $k,\ell$, $\phi(n,k,\ell)$ is the least number of distinct distances spanned by a set of $n$ points in the plane in which every $k$ points determine at least $\ell$ distinct distances. The survey reads $\phi(n,3,3)$ as the least number of distinct distances among $n$ points spanning no isosceles triangle, degenerate (collinear) isosceles triangles included.

What the survey records (p. 13):

• Lower bound $\phi(n,3,3)=\Omega(n)$: with no isosceles triangle, each point has $n-1$ distinct distances to the others.
• Upper bound $\phi(n,3,3)<n2^{O(\sqrt{\log n})}$, observed by Erdős: take Behrend's set $a_1<\cdots<a_n$ of positive integers with no three-term arithmetic progression and $a_n<n2^{O(\sqrt{\log n})}$, and place the points $(a_i,0)$ on a line; they span no isosceles triangle and at most $a_n$ distances.
• Erdős conjectured $\phi(n,3,3)=\omega(n)$.

Problem 28 (p. 13). "Find the asymptotic value of $\phi(n,3,3)$." (quoted)

Read depth

Claims checked on the print. Behrend's construction is reported as the survey states it and was not checked against its source here.

Bears on

• Problem 657: the problem asks whether $n$ planar points with no isosceles triangle must determine at least $f(n)n$ distinct distances for some $f(n)\to\infty$, that is, Erdős's conjecture $\phi(n,3,3)=\omega(n)$ as the survey states it. The survey records the bounds $\Omega(n)$ and $n2^{O(\sqrt{\log n})}$ and leaves the problem open as Problem 28.