Result 166, Combinatorics

The higher-dimensional Erdős distinct-distances conjecture

For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least cdn2/dc_dn^{2/d} distinct distances, with cd>0c_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.

Proof

The bigger picture

Why it matters

How few different distances can a large collection of points determine? This manuscript claims a sharp answer in every fixed dimension at least three, identifying how much variety in lengths any arrangement must contain.

What changes?

The manuscript reports that, for every fixed integer dimension d at least three, any set of n distinct points, with n at least two, in ordinary Euclidean space determines at least c_d times n raised to the power 2/d different distances. Here distances are straight-line lengths between pairs of points, counted without repetition. The positive constant c_d depends only on the dimension, not on the number or arrangement of points.

What does that help mathematicians do?

In three dimensions, for example, the claimed minimum grows like n to the two-thirds power, up to constant factors. The integer grid achieves the same order, so no larger power of n can hold universally. Researchers could therefore use the bound to rule out proposed configurations with too few distance values, while treating grids as examples of the correct scale for the minimum.

Are there practical applications?

Its immediate value is foundational: it would give combinatorial geometry a sharp baseline for comparing how point arrangements reuse the same lengths. This is a statement about what configurations can exist, not a method for constructing them or computing their distances faster. The supplied material describes no specific practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The higher-dimensional Erdős distinct-distances conjecture

September 23, 2026 103 pages

For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝd determines at least cdn2/dc_d n^{2/d} distinct distances, where cd>0c_d\gt 0 depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.