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The higher-dimensional Erdős distinct-distances conjecture
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Abstract
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.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Thomas Bloom (erdosproblems.com blog), Erdős problems in the OpenAI proofs (blog post, 8 October 2026). In the section on Erdős Problem 1083, Bloom says the claimed bound is "actually stronger than Erdős' conjecture, which allowed for an $n^{o(1)}$ loss". He also notes that the proof "has not been formalised", "appears to be heavily algebraic" and uses many ideas from Tidor–Yu–Zakharov.
Formalization
- Yao Xu (@cadamcat), distinct-distances-lean4 (GitHub repository, first commit 10 October 2026). An independent Lean 4 development, "developed with AI assistance" per its README, that says it follows this paper's proof of Theorem 1.1 and also proves the matching grid upper bound; the README does not report the result of its axiom-check script.
- mo271, ErdosProblems/1083:
erdos_1083is true (formal-conjectures issue #6905, opened 7 October 2026). The issue cites this paper and proposes resolving the Formal Conjectures statementerdos_1083with it as the source; it is open with the label "solution found" and notes that the release includes no Lean proof.
Related prior literature mentioned by others
- Jonathan Tidor, Hung-Hsun Hans Yu, Dmitrii Zakharov, The Erdős distinct distances problem in ℝ³ (arXiv:2608.14454, 14 August 2026). This paper proves $N^{2/3-o(1)}$ distinct distances in ℝ³, and Bloom's blog post names it as the $d=3$ antecedent whose ideas the paper reuses; it is already in the paper's bibliography.
- János Pach, Guth and Katz’s Solution of Erdős’s Distinct Distances Problem (Combinatorics and more, 2010-11-20). Kalai's October 2026 post on this paper points to this account of the planar Guth–Katz bound and the Elekes–Sharir approach, background for the paper's planar induction base and rigid-motion methods.
Possibly relevant literature (found by the agent, not checked)
- Adam Sheffer, Distinct Distances: Open Problems and Current Bounds (arXiv:1406.1949, v3 2 July 2018). A survey of distinct-distances variants and their best known bounds, relevant to the paper's account of prior bounds and conjectures for $d\ge3$; not in the paper's bibliography.
News and blogs
- Gil Kalai, Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans (Combinatorics and more, 2026-10-07). Item 8 links this paper's sharp-order distinct-distances claim and connects it to earlier blog discussions, in a post stating that the release's proofs require community verification.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 166
Independent sites; their content is not reviewed here.
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}
}Family 166: The higher-dimensional Erdős distinct-distances conjecture
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. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
This is the only paper in this family.