Stanley's Claw-Free Schur-Positivity Conjecture

Precise statement

Is the chromatic symmetric function $X_G$ Schur positive for every claw-free graph $G$? Two explicit $12$-vertex line graphs have Schur coefficients $-64$ and $-40$ at $s_{(3,3,3,3)}$.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

ChatGPT-5.6 Sol Pro

Both counterexample graphs were found with ChatGPT-5.6 Sol Pro; the negative coefficients were confirmed by independent computations.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Author preprint with independent computational checks of the negative Schur coefficients. Not yet peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-07-23

   arXiv:2607.21508 - Chromatic symmetric functions of claw-free graphs are not Schur positive

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

construction (source-reported)

Source-reported tools: construction.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.