The proper hat-guessing number of
Determine the exact proper hat-guessing number of the complete graph on six vertices with one edge removed. The known general bounds leave .
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Graph theory; hat-guessing games; permutation groups
- Posed by
- Adriaensen et al.
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-03
- Model
- GPT-5.6 Pro
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 7 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
We prove . The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The release also classifies all repairable orbit labels and proves an even- obstruction for set-symmetric line-permutation twin rules. It does not solve the general family.
What the AI did
OpenAI GPT-5.6 Pro contributed substantively to literature search, target selection, construction search, proof development, code generation, exact verification, adversarial critique, and manuscript preparation. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, determined the claim scope, approved disclosure, and accepts responsibility.
Verification
A separate adversarial technical review independently regenerated the principal finite core and prompted the scope and proof-presentation corrections incorporated into this public version. The dependency-free verifier regenerates the group, repairs, coordinate-line checks, label classification, Witt-design check, and residual right-degree census. Conventional journal peer review and Lean verification are not claimed.
Sources
Submitted by Matthew Protti on