Not evaluated
Hardness of coloring three-colorable graphs (family 106)
The gist
In its release of October 6, 2026, OpenAI claims that finding large independent sets in graphs whose vertices can be coloured with three colours is computationally hard.
DiscoverIA status on October 12, 2026: not evaluated, meaning no third-party evaluation found.
AI draft, not reviewedWhat has not been reviewed?
Text written by an AI (DiscoverIA pipeline), on . The status and its sources come from the registry; only the explanatory text has not been reviewed. A reported error is corrected, then the text is reviewed.
Cite this entry
A single status snapshot since Oct 12, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-106-hardness-coloring-three-colorable-graphs-r1
@misc{discoveria-oai-fam-106-hardness-coloring-three-colorable-graphs-r1,
author = {{DiscoverIA}},
title = {Hardness of coloring three-colorable graphs (family 106)},
howpublished = {DiscoverIA registry, entry oai-fam-106-hardness-coloring-three-colorable-graphs},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-106-hardness-coloring-three-colorable-graphs#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 20 s reading time, estimated
What it changes
If the manuscript held, finding a large set of vertices, no two of which are linked (an independent set), would be computationally hard. This would hold even for graphs whose vertices can be coloured with three colours so that no two linked vertices share a colour.
What remains unclear
No independent assessment is recorded, and the sources read do not say under which assumptions this hardness is stated. The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI and marked “review: status: unchecked” in formalization.yaml.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 106.md
Step 1 of 4 on the scale: not evaluated. Next step: under review.
- Announced
- Recordednot evaluated
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
The result is declared neither right nor wrong.
To move to “Under review”: a public review under way. To “Peer reviewed”: a journal publication or a review by named experts.
Sources
- Main sourcegithub.com · CONTENTS.mdlink only, reading not recorded
- Repositorygithub.com · Hardness-of-finding-large…link only, reading not recorded
- Lean proofgithub.com · 106.mdpage read word for word
- Lean proofgithub.com · IndependentSets.leanlink only, reading not recorded
Method
OpenAI publishes one manuscript, “Hardness of finding large independent sets in three-colorable graphs”, dated September 24, 2026. The repository also holds lean/docs/106.md, which describes the scope of the Lean formalisation (1 Comparator statement), and lean/ComparatorChallenges/IndependentSets.lean. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 12, 2026 from github.com (page read, quotation checked word for word: “describes the scope of the Lean formalization”).
Limits
No named third party: the recorded checks come from the claimant or from a platform tool. The claimant supplies a Lean proof; as of this date, DiscoverIA has found no independent verification of this proof or of the result.
Key facts
- Claimed by
- OpenAI (unreleased internal model)
announced on Oct 6, 2026 - Nature
- new result
- Field
- Theoretical computer science
- In openai/math
- family 106, section “Theoretical computer science”, 1 manuscript dated Sep 24, 2026
- Lean proof
- supplied by the claimant, marked “unchecked” (not reviewed) in its repository
- Revisions
- none reported by the claimant
- Checked by
- claimant (Lean supplied, not checked by a third party)
- Sources
- 4 : 1 read word for word, 3 links only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
Family with a Lean proof supplied by the claimant. No third-party check read; the source marks this proof as not reviewed (“unchecked”).
Source: github.comread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 103 of 372, by family number
- Not evaluated : Derandomization of logarithmic space: L=RL=BPL (family 103)
- Not evaluated : Quasipolynomial algorithms for mean-payoff, stochastic and parity games (family 104)
- Not evaluated : The 2-to-1 Games Conjecture with perfect completeness (family 105)
- Not evaluated : Hardness of coloring three-colorable graphs (family 106)
- Not evaluated : Matrix multiplication exponent at most 9/4 (family 107)
- Not evaluated : A cubic permanent–determinant lower bound (family 108)
- Not evaluated : Integer multiplication below n log n (family 109)
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