---
title: "The higher-dimensional Erdős distinct-distances conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 166; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "combinatorics"
  - "unformalized"
seed_rank: 2021
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "The higher-dimensional Erdős distinct-distances conjecture"
    url: "https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The higher-dimensional Erdős distinct-distances conjecture

## One-sentence takeaway

OpenAI's result family 166 (Combinatorics) claims: For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least $c_dn^{2/d}$ distinct distances, with $c_d\gt 0$ depending only on dimension.

## Why it matters here

Combinatorics is where the library's procedural-generation, graph and grid work (GRID COMMAND maps, tilings, WFC) gets its theory. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
- *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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [The higher-dimensional Erdős distinct-distances conjecture](https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math