The higher-dimensional Erdős distinct-distances conjecture

For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝ^d determine at least c_dn^{2/d} distinct distances, with c_d\gt 0 depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

Manuscript

The higher-dimensional Erdős distinct-distances conjecture

September 23, 2026 · source · cite as \cite{OAI:The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026}

For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝ^d determines at least c_d n^{2/d} distinct distances, where c_d\gt 0 depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.

Result 166 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Agora members.