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The proper hat-guessing number of K5−eK_5-e

Determine the exact proper hat-guessing number of the complete graph on five vertices with one edge removed. The existing bounds left HGP(K5−e)\mathrm{HG}_P(K_5-e) in {7,8}\{7,8\}.

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Construction
Field
Graph theory; hat-guessing games
Posed by
Adriaensen et al.
Year posed
2026
Years open
0y
Solved
2026-09-02
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

The exact value is HGP(K5−e)=8\mathrm{HG}_P(K_5-e)=8. The lower bound uses explicit legal twin-player rules over F23\mathbb F_2^3. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion lemma for Kn−eK_n-e. It does not solve the full Kn−eK_n-e family or determine HGP(C5)\mathrm{HG}_P(C_5).

What the AI did

OpenAI GPT-5.6 Pro contributed substantively to literature search, problem selection, construction search, proof development, code generation, computational verification, adversarial critique, and manuscript preparation. The central F23\mathbb F_2^3 construction and Hall-completion proof were developed in a model-assisted process. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, set the scope, approved disclosure, and accepts responsibility.

Verification

No independent mathematical review yet. The public disclosure contains a self-contained proof, a seven-entry finite certificate, and a dependency-free Python verifier. The verifier has been run successfully from a clean clone and from the v0.1 release archive; it reconstructs the residual incidence graph, a saturating matching, and a complete 6,720-entry strategy, then checks all 8,400 proper colorings. This remains author-controlled verification.

Sources

Submitted by LuckyHawk816 on

Changelog2 changes

Discussion