OPEN

Is it true that if $A\subset \mathbb{R}^2$ is a set of $n$ points such that every subset of $4$ points determines at least $5$ distances then $A$ must determine $\gg n^2$ distances?

A problem of Erdős and Gyárfás. Erdős could not even prove that the number of distances is at least $f(n)n$ where $f(n)\to \infty$.

More generally, one can ask how many distances $A$ must determine if every set of $p$ points determines at least $q$ points.

See also [657].