The unit distance problem. In \cite{Er94b} Erd\H{o}s dates this conjecture to 1946. In \cite{Er82e} he offers \$300 for the upper bound $n^{1+o(1)}$.

This would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(n^{4/3})$, due to Spencer, Szemer\'{e}di, and Trotter \cite{SST84}. In \cite{Er83c} and \cite{Er85} Erd\H{o}s offers \$250 for an upper bound of the form $n^{1+o(1)}$.

Part of the difficulty of this problem is explained by a result of Valtr (see \cite{Sz16}), who constructed a metric on $\mathbb{R}^2$ and a set of $n$ points with $\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemer\'{e}di, and Trotter \cite{SST84} generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.

See a survey by Szemer\'{e}di \cite{Sz16} for further background and related results.

See also [92], [96], [605], and [956]. The higher dimensional generalisation is [1085].