Weak unit distance conjecture

For a finite point set $P\subseteq\mathbb R^2$, consider the pairs of points at Euclidean distance $1$.

Weak unit distance conjecture. For every $\varepsilon>0$, there exists a constant $C_\varepsilon>0$ such that every finite point set $P\subseteq\mathbb R^2$ determines at most

$$C_\varepsilon |P|^{1+\varepsilon}$$

unit distances.

The source attributes this conjecture to Erdős and records a best known bound due to Spencer, Szemerédi and Trotter. Its resolution status is not supplied.