A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz \cite{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 \cite{Er75f} Erd\H{o}s 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.


References


[Er75f] Erd\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.

[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erd\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.