Breakthrough Tracker record

Erdős Problem #90: the planar unit-distance conjecture is false

Arbitrarily large planar point sets can determine at least n^(1+c) unit-distance pairs for an absolute c greater than zero, contradicting the proposed n^(1+o(1)) growth. A companion paper gives an explicit exponent greater than 1.014.

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Stable ID: math-erdos-90-unit-distance-2026
Revision: math-erdos-90-unit-distance-2026.v1
Field: Mathematics · Discrete geometry
Evidence: Tier 1 · Peer reviewed: No
Record state: Current · Human-verified disproof in expert-authored preprints
Last checked: 2026-07-29

AI role

An OpenAI model generated the construction and proof; a group of mathematicians produced a shorter human-verified account.

Record details

Problem or result: Erdős Problem #90, the planar unit-distance conjecture
Authors: Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood
Institutions: Multi-institution collaboration
Result date: Announced and submitted May 20, 2026

Why it matters

The disproved near-linear-growth conjecture had guided the unit-distance problem for decades.

Limits

The extremal unit-distance problem remains open. A large gap persists between the new lower bound and the classical O(n^(4/3)) upper bound.

Sources

Primary: OpenAI research announcement
Primary: Original proof PDF
Primary: Alon et al., Remarks on the disproof of the unit distance conjecture
Primary: Will Sawin, An explicit lower bound for the unit distance problem
Independent: Erdős Problems #90
Independent: Nature news coverage

Correction and revision history

Added on 2026-07-22 as a human-verified disproof while preserving that the broader extremal problem remains unsolved.

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