The higher-dimensional Erdős distinct-distances conjecture

What OpenAI claims

For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least c_dn^2/d distinct distances, with c_d > 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.

Paper

1. The higher-dimensional Erdős distinct-distances conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

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

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.