What was already known
Voss, building on Larson, forced an odd cycle with at least two chords in every K₄-free 4-chromatic graph. Erdős asked whether stronger local 3-colorability conditions forced more chords.
AI involvement
An internal OpenAI model found the pentagon-block graph family and an original color-criticality proof. Alexeev, Putterman, Sawhney, Sellke and Valiant say they verified the model solution and replaced part of its proof with their stronger 2-degeneracy argument.
The model supplied the central graph family and original proof; task prompting is not documented finely enough for AI-autonomous.
Importance
The bounded-chord family answers the specific Erdős graph-coloring question.
Limitations
The paper’s stronger proof that every proper subgraph is 2-degenerate is due to the human coauthors. No separate machine certificate for the family is documented.
Technical verification
The five human coauthors report that they verified the model’s graph solution before publication. Their later stronger degeneracy argument is credited to them.
Independent checking
human verification while digesting the proof; retained proof also uses their 2-degeneracy argument; §1, Comment on the use of AI; §4, Theorem 4.1 and Propositions 4.5–4.10
- Checker
- Alexeev, Putterman, Sawhney, Sellke and Valiant
- Scope
- Model's Problem 1091 graph family and original color-criticality solution
- Basis
- Reported by the linked checker
Human understanding
The authors explain the pentagon forcing and chord count and supplied the simpler stronger degeneracy route; this establishes understanding of the central construction.
Correction and publication history
The bounded-chord family answers the specific Erdős graph-coloring question.
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