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.

References

No one knows how to improve the exponent 4/3, even for uniformly spaced sets.

— Perspectives on the unit distance problem  (2609.10791 - Guth, 9 Sep 2026) in Section 5, subsection “Why is the exponent 4/3 hard to improve?”

The Erd\H{os unit distance conjecture} states that $U(t) = O(t{1 + \varepsilon})$ for every $\varepsilon > 0$.

— On the shatter function of semilinear set systems  (2501.10032 - Basit et al., 17 Jan 2025) in Section 1, Introduction