Erdős unit-distance problem and square-grid optimality conjecture

Erdős posed the planar unit-distance problem in 1946, conjecturing that the maximum number of unit-distance pairs among $n$ points is $n^{1+o(1)}$. The new work claims this square-grid optimality conjecture is false, without determining the true extremal order.

Known results

• The rescaled square grid gives $n^{1+\Omega(1/\log\log n)}$ unit distances (Erdős, 1946).
• The best general upper bound is $O(n^{4/3})$ (Spencer–Szemerédi–Trotter, 1984).

May–August 2026 counterexample

An internal OpenAI model reportedly generated a proof producing infinitely many planar configurations with at least $n^{1+\varepsilon}$ unit distances. Human mathematicians simplified and improved it; later work gives an explicit exponent about $1.014114$. These technical write-ups corroborate the claimed refutation of square-grid optimality, but the exact extremal function remains open.

Current status (as of August 2026): The $n^{1+o(1)}$ square-grid optimality conjecture is supported as refuted, while the exact planar unit-distance extremal order between the known lower and upper bounds remains open.