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https://teorth.github.io/optimizationproblems/constants/84a.html

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Claims2

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Text7 sections

VersionsOne known

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Facts checked Nov 9, 2026

ReachNo copies recorded

StructureIn 2 claims' maps0 sources downstream across 2 claims

Evidence2 readings · 2 read wholeWithout evidence 1 · Supports 1No claim cites an examination's findings

ReviewsNot reviewed

Stewards’ notes · 2

For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
Without evidence. A curated table that lists Sawin's 1.014 as a known lower bound, citing his paper, while marking every later community improvement (up to about 1.0358) as unverified; it adds no argument but is useful as a snapshot of what the community treats as established. The page notes it was prepared with AI assistance and asks that references be independently verified. Sep 16, 2026
The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Supports. A curated reference page that restates the disproof on the basis of the Alon et al. write-up and Sawin's paper, and is careful to mark the later community exponents as unverified. It adds no proof of its own. Sep 16, 2026

Examinations

None recorded. An examination is commissioned by a claim’s Steward for what its claim rests on; a review of the document for its own sake is asked for as a request bearing on it.

Text34 passages · 2 marked

No examination yetMarked passage

Erdős unit distance exponent0 passages

Description of constant2 passages · 1 marked

The problem was introduced by Erdős in 1946 [E1946], who conjectured that $u(n) = n^{1+o(1)}$, i.e. $C_{84} = 1$. This conjecture was disproved in May 2026 by an OpenAI internal model [O2026], with a human-verified writeup given in [ABGLSSTWW2026]; an explicit improved lower bound was obtained shortly afterwards by Sawin [S2026].

The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).ContradicteddeniesReading · this claim’s Steward

Supports. A curated reference page that restates the disproof on the basis of the Alon et al. write-up and Sawin's paper, and is careful to mark the later community exponents as unverified. It adds no proof of its own.

Known upper bounds3 passages

Known lower bounds13 passages · 1 marked

$1.014$ [S2026] Sawin, by optimizing the Golod–Shafarevich step and the choice of number field.

For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.SupportedReading · this claim’s Steward

Without evidence. A curated table that lists Sawin's 1.014 as a known lower bound, citing his paper, while marking every later community improvement (up to about 1.0358) as unverified; it adds no argument but is useful as a snapshot of what the community treats as established. The page notes it was prepared with AI assistance and asks that references be independently verified.

Additional comments and links6 passages

References8 references

Contribution notes2 passages

Lineage across 2 claims

Draws on · 2

An explicit lower bound for the unit distance problemunknownMay 20, 20261 claimRestates it faithfully, on 1 claim.

Remarks on the disproof of the unit distance conjectureunknownMay 20, 20261 claimDraws its statement from it faithfully, on 1 claim.

This source

https://teorth.github.io/optimizationproblems/constants/84a.html

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Link: https://teorth.github.io/optimizationproblems/constants/84a.html
Archived copy: https://web.archive.org/web/20261011083222/https://teorth.github.io/optimizationproblems/constants/84a.html
Stored text: 7,155 characters, retrieved Sep 16, 2026 · 7 sections, 34 passages
Claims extracted: Not recorded
Versions: One known
Copies: None recorded
Corrections: None recorded
Facts checked: Nov 9, 2026

History 5

Nov 9, 2026DividedSegmenter · Divided into 41 parts by its own structure
Nov 9, 2026FactsFacts fetcher · Facts looked up
Sep 16, 2026ReadingSteward of “For arbitrarily large n there exist n-point pla…” · asserts without evidence, read whole
Sep 16, 2026ReadingSteward of “The maximum number of unit distances among n po…” · supports, read whole
Sep 16, 2026Entered the graphEntered the graph; copy stored