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

Sawin's explicit exponent. The constructive disproof can be realised with $c \ge 0.014114$ (absorbing the implicit constant $C$ of Sawin's Theorem 1 into a slightly smaller exponent for all large enough $n$). Reference: Theorem 1 of Sawin, arXiv:2605.20579 (2026).