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Source. Problem 31, p. 14 (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). $\phi(n,k,\ell)$ 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.

What the survey records (p. 14 and Table 3):

• Erdős asked whether $\phi(n,4,5)=\Theta(n^2)$.
• The best lower bound is $\phi(n,4,5)=\Omega(n)$: in such a set, a circle centred at a point of the set meets at most two points of the set.
• Table 3 lists the trivial upper bound $O(n^2)$.

Problem 31 (p. 14). "Find the asymptotic value of $\phi(n,4,5)$." (quoted)

The survey calls this case one of the main variants of the problem, about which not much is known (p. 14).

Read depth

Claims checked on the print.

Bears on

• Problem 135: the problem asks whether $n$ planar points of which every four determine at least five distances must determine $\gg n^2$ distances, that is, Erdős's question whether $\phi(n,4,5)=\Theta(n^2)$ as the survey states it. The survey records only the lower bound $\Omega(n)$ and leaves the problem open as Problem 31.