The higher-dimensional Erdős distinct-distances conjecture
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 \(c_d\gt 0\) depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.
Cite
@misc{OAI:The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026,
author = {{OpenAI}},
title = {{The higher-dimensional Erd{\H{o}}s distinct-distances conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf}{OAI:The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026}},
year = {2026}
}