Score-Determined Induced Tournament Statistics: an All-Orders Classification
A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different.
Question: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size?
Answer: Exactly the linear combinations of degree-multiplicity counts , where counts how many of the chosen vertices have exactly internal wins. These functions satisfy one linear relation, so score-determined statistics have dimension .
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Computation
- Field
- Tournament theory
- Posed by
- —
- Year posed
- —
- Years open
- —
- Solved
- 2026-08-14
- Model
- GPT Sol 5.6
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 3 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
Absolutely everything, i occasionally steered and redirected.
Verification
Checked by this site on 17 August 2026: the Zenodo reproducibility archive (21959291, Howard-Stanley) is real and carries the manuscript and code. The classification itself was not checked here, no independent review exists, and - flagged plainly - nobody has swept the tournament-theory literature for prior appearances of this question, which for a self-posed problem is the main risk. The disclosure ("absolutely everything, I occasionally steered") is the frankest ai-discovered claim in the catalog.
Source
Submitted by shemshallah on