What was already known
Wegner's 1965 conjecture bounded the piercing number by 2ν−1. Earlier rectangle examples reached 5/3, while a diagonal-line construction approached ratio 2 without violating the conjecture; earlier clique-LP lower bounds reached approximately 2.
AI involvement
GPT-5.5 Pro helped search for rectangle counterexamples, including search code. Ajwani, Gajjala, Raman and Ray distilled the promising constructions and wrote and checked the final proofs.
The documented model-run search produced promising rectangle counterexamples that the authors distilled into the central construction; the final proof and generalization remain credited to the authors.
Importance
Disproves Wegner’s rectangle-piercing bound and raises the known lower ratio toward 5/2.
Limitations
The separate LP upper bound and chromatic construction are additional results of the authors. The paper does not identify which final coordinates or parameters first appeared in model output.
Technical verification
All mathematical proofs were reviewed and rewritten by the four human authors, including the explicit counterexample and grid-package generalization. This is an author-reported complete human proof review, not a reproduced check. Auxiliary integer programs and Lean work have no stated theorem-matching scope here.
Formal checking
All mathematical proofs were reviewed and rewritten by the four human authors, including the explicit counterexample and grid-package generalization. This is an author-reported complete human proof review, not a reproduced check. Auxiliary integer programs and Lean work have no stated theorem-matching scope here.
- Checker
- Ajwani, Gajjala, Raman and Ray, the paper's human authors
- Scope
- The rectangle counterexample and generalized package construction disproving Wegner's bound
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the eight-rectangle package, grid composition and port-count argument, including why the family is triangle-free and has packing number 16.
Correction and publication history
The AI search supplied promising counterexamples; the four authors distilled the construction and checked the proofs.
Read this published revision