The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique
The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all there is an such that every finite point set of at least points contains collinear points or points that pairwise see each other. True for , , the first case left open: every finite point set of size at least has four collinear points or six pairwise visible points.
- Result
- Proved
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Discrete geometry
- Posed by
- Jan Kára, Attila Pór and David R. Wood
- Year posed
- 2005
- Years open
- 21y
- Solved
- 2026-08-19
- Model
- GPT-5.6 Sol Pro
- Vendor
- OpenAI
- Collaborators
- Édouard Bonnet
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper's disclosure, in full: "After one fruitless attempt and the now customary generic encouragement, a relatively detailed proof of Theorem 1 was provided by GPT-5.6 Sol Pro after pondering for 222 minutes. The author's contributions were limited to checking the proof, simplifying some parts, and eventually writing the paper in a way that would give the author (and hopefully other human readers) a more pleasant reading experience."
Verification
Checked by this site on 21 August 2026 against the paper (arXiv:2608.19468v1, 7pp): the disclosure is verbatim as quoted, the conjecture's 2005 attribution is in the text, and the paper states this is the first open case resolved since 2009. The proof was not checked here. The threshold is astronomical but the statement is unconditional. Days-old preprint, no independent review.