The proper hat-guessing number of
Determine the exact proper hat-guessing number of the complete graph on eight vertices with one edge removed. The general bounds leave .
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI co-developed
- Method
- Computation
- Field
- Graph theory; hat-guessing games; finite geometry; matching theory
- Posed by
- Adriaensen et al.
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-04
- 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 using an explicit -equivariant strategy. A compact formula fixes 990 normalized twin decisions, and a 48,510-edge residual-orbit matching induces consistent rules on 53,460 labelled clique-view orbits. Independent implementations regenerate the certificate and check all 138,378,240 proper fourteen-colourings with zero failures. Unlike the constructions, residual right degrees reach eight, so the explicit global matching is load-bearing. A separately labelled companion exhibits one fixed clique completion compatible with equivariant twin-rule pairs and proves 380 optimal within the stated independent whole-tail reversal model; that companion's independent review is pending. The release does not solve the general family or .
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, certificate design, and manuscript and release preparation. The equivariant construction, residual-matching proof, Hall-obstruction analysis, and shared-completion family were developed in a human-directed model-assisted research process. Matthew Protti selected and framed the target, directed and evaluated the work, commissioned review, required exact checks, approved the public scope, and accepts responsibility.
Verification
Unreviewed on this site's ladder. The frozen commit bacb79e3 carries STATUS.json with headline status PROVED_COMPUTER_ASSISTED and project review ACCEPTED, an AI_USE_AND_PROVENANCE.md naming the model and the division of labour, and a C++ sweep of all 138,378,240 proper fourteen-colourings with zero coverage failures across 48,510 residual orbits. The Gate 2/3 Hall-obstruction archive was reviewed as ACCEPT_WITH_EXPLICIT_REPAIRS and the repairs applied. The 2^380 shared-completion companion is a separate claim whose independent review is still pending and is not part of this entry. As with the siblings: a real adversarial review of an artefact, not a named expert or a proof assistant, and this site has not rebuilt it.
Sources
- PaperK8-e theorem and proof
- CodeK8-e public research disclosure v0.1 (GitHub)Verified construction and full-colouring checker
- AnnouncementPublic draft PR #3 and review surface
- Problem recordHat guessing with proper colorings (Adriaensen et al.)
- OtherReview-pending shared-completion companionRepaired Gate 2/3 Hall-obstruction archive
Submitted by Matthew Protti on