Erdős unit-distance conjecture

Prove that the maximum number U(t) of pairs of points at unit distance among all sets of t points in the plane satisfies U(t)=O(t^{1+\varepsilon}) for every \varepsilon>0.

Background

The paper relates the shatter function of the semilinear-looking set system of unit-distance pairs in the plane to the extremal function U(t), which counts the maximum number of unit-distance pairs determined by t planar points. The authors record the best-known lower and upper bounds and state the Erdős unit-distance conjecture as the conjectured near-linear upper bound. They further note that, if the conjecture holds, the associated shatter function is not asymptotic to a real power function.