Problem #89: Does every set of $n$ distinct points in $\mathbb{R}^2$...

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Progress

A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz [GuKa15] who proved that there are always $\gg n/\log n$ many distinct distances.

A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct distances, or even $\gg n$ many such points, or even that this is true averaged over all points - for example, if $d(x)$ counts the number of distinct distances from $x$ then in [Er75f] Erdős conjectured\[\sum_{x\in A}d(x) \gg \frac{n^2}{\sqrt{\log n}},\]where $A\subset \mathbb{R}^2$ is any set of $n$ points.

See also [661], and [1083] for the generalisation to higher dimensions.