Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.

Constructive form of the disproof. There is an absolute constant $c > 0$ such that infinitely many $n$ admit a configuration realising at least $n^{1+c}$ unit distances.

This is the qualitative content of Theorem 1.1 of Alon–Bloom–Gowers–Litt–Sawin–Shankar– Tsimerman–Wang–Matchett Wood, *Remarks on the disproof of the unit distance conjecture* (2026). An explicit bound $c \ge 0.014114$ is given by Sawin, *An explicit lower bound for the unit distance problem* (2026); see erdos_90.variants.sawin_explicit below.