Weak pinned distance conjecture

For a finite point set $P\subseteq\mathbb R^2$, a point $q\in P$ determines the distinct Euclidean distances from $q$ to the other points of $P$.

Weak pinned distance conjecture. For every $\varepsilon>0$, there exists a constant $C_\varepsilon>0$ such that every finite point set $P\subseteq\mathbb R^2$ contains a point $q\in P$ determining at least

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

distances to the other points of $P$.

The conjecture is attributed in the source to Erdős. It is presented as an open conjecture and is used to derive a graphical distinct-realisation conjecture.