What was already known
The source gives published upper bounds R₄(3)≤62 and S(6)≤1836. The stronger R₄(3)≤61 is a separate computer-assisted claim by M. Tatarevic and is not proved in this repository.
AI involvement
Adam McKenna directed the project. The README explicitly credits ChatGPT agents with first proposing the centred bound and frontier arguments, and Claude with checking the steps and writing the Lean proofs.
The pinned README credits ChatGPT agents with first proposing the centred-bound and frontier arguments, while Claude checked and formalized them under Adam McKenna’s direction.
Importance
Derives a sharper conditional Schur bound and constraints on a hypothetical extremal colouring from a general Ramsey-to-Schur inequality.
Limitations
This is a conditional theorem: the Lean code takes R₄(3)≤61 as a hypothesis and does not establish it, nor does it prove S(6)≤1800 or exclude a colouring at 1801. The author says the centred-bound method may be older; full texts of two potentially relevant earlier works were not reviewed, so novelty of that general method is unresolved. Separate LRAT checks of Tatarevic’s SAT instances do not import the Ramsey hypothesis into this Lean theorem. No human peer review is reported.
Technical verification
The source reports all 14 compared theorems proved with standard Lean axioms and CI Comparator checks. The claim assessed here is conditional; the R₄(3)≤61 premise is not proved or imported by this repository.
Formal checking
The source reports all 14 selected theorem proofs use only standard Lean axioms, with CI Comparator statement checking; the R₄(3)≤61 premise remains an explicit Lean argument.
- Checker
- Lean 4 kernel and documented comparator/certificate checks run by the repository authors
- Scope
- If every four-colouring of the edges of K₆₁ contains a monochromatic triangle, equivalently R₄(3)≤61, then S(6)≤1801. More generally, if R_k(3)≤r, then S(k+1)≤2(k+1)⌊(r−1)/2⌋+1. The Lean proof also gives structural restrictions for a six-colour sum-free colouring at the 1801 frontier under the same R₄(3) hypothesis.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The source documents AI audit and formal proof but no independent human proof review or human explanation of the AI-proposed argument. Few is the evidence default.
Correction and publication history
Primary-source review of the exact conditional theorem, stronger published baselines, AI-origin arguments, unproved Ramsey premise and prior-method uncertainty.
Read this published revision