What was already known
Perfect divisibility was known to imply the hereditary chromatic bound. Sivaraman asked whether the converse holds.
AI involvement
Wang and Xia credit ChatGPT with identifying P(17) during exploration; they checked it and developed the proof. The search transcript is not published.
The authors explicitly attribute discovery of the decisive P(17) counterexample to exploratory ChatGPT use, while the full proof and extension are human work.
Importance
The explicit P(17) example settles the proposed converse and gives an infinite counterexample family.
Limitations
Wang and Xia supplied the proof and join extension. Their separate chair-free results are outside the model-found construction. The finite code and checks are reported by the authors.
Technical verification
The authors explicitly report checking all mathematical arguments and computational results, and their released Python program exactly enumerates the finite graph cases used in the proof. These complete human and exact-computation checks are source-reported.
Formal checking
Wang and Xia report checking the finite graph proof and computational enumeration of P(17).
- Checker
- Zhiyu Wang and Weihao Xia, paper authors
- Scope
- Theorem 1.1: P(17) meets the hereditary chromatic bound but is not perfectly divisible
- Basis
- Reported by the linked checker
Formal checking
The cited Python program exactly enumerates P(17) invariants and the 51 relevant triangle-free eight-vertex subsets; Wang and Xia report checking its results.
- Checker
- Pinned verify_paley17.py program, with results checked by Wang and Xia
- Scope
- Finite P(17) invariants and enumeration of 51 triangle-free eight-vertex subsets in the proof of Theorem 1.1
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors explain why all induced subgraphs obey the bound and why the only candidate perfect-division complements contain induced five-cycles.
Correction and publication history
The explicit P(17) example settles the proposed converse and gives an infinite counterexample family.
Read this published revision