Author verified AI-led

First open case of the big-line-big-clique conjecture

Every finite point set of size at least $10^{11055931}$ contains four collinear points or six points that pairwise see each other, settling the first open case of the Kára-Pór-Wood conjecture.

Lab: Independent
Model: GPT-5.6 Sol Pro
Field: Mathematics
Date: 2026-08-19
Human collaborators: Édouard Bonnet
Problem posed: 2005 · open 21 yrs

Sources

Original work

arXivlarge finite point sets have 4 collinear points or a 6-clique↗

Independent commentary

VibeMathedthe Kára-Pór-Wood big-line-big-clique conjecture↗arXivCambie, Four collinear points or six visible points↗

What was found

The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k$ and $l$ there is an $n$ such that every finite point set of at least $n$ points contains $l$ collinear points or $k$ points that pairwise see each other, where two points see each other if no other point of the set lies between them. The paper proves the case $l = 4$ , $k = 6$ , the first case left open, with the explicit and very large threshold $10^{11055931}$ . The proof combines a lower bound on distance to 5-colorability, a Turán-type stability and edge-density argument, and matching harmonic upper and lower bounds.

Novelty check

The conjecture is named and dated to Kára, Pór and Wood in 2005, and the paper identifies $l = 4$ , $k = 6$ as the first case left open by the existing partial results, which fixes the prior state of the art. The threshold is explicit, so the statement is falsifiable as written rather than asymptotic hand-waving. No prior proof of this case appears in the discrete geometry literature or in the vibemathed record.

Caveats and known objections

A short preprint, unrefereed, not formalized, with no independent check on record, and vibemathed lists it as a candidate pending review. The threshold is astronomically large and the result settles one case rather than the conjecture. Stijn Cambie has since brought the threshold for the same case down to 880 points in a three-page note, itself AI-assisted and marked not for publication. Autonomy is graded ai-led on the paper's own AI disclosure, which is unusually specific about the sequence: after one fruitless attempt and what the author calls 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 open problem

MathDBBig line big clique conjecture

315965

Nothing mechanically. It records what a problem says, what is known about it, and the standing of any claimed solution.

Tracks the same Bonnet preprint resolving the (6, 4) case that this entry records.

Also recorded at

vibemathedbig-line-big-clique-four-collinear-or-six-clique

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

Nobody outside the lab has checked this yet.

Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.

Submit a check Challenge the grade

Or on GitHub: submit a check challenge the grade send a correction or send a pull request

History of this entry

2026-09-24CorrectedRecorded Stijn Cambie's improvement of the threshold for the same case to 880 points. No grade moved. source ↗
2026-09-10CorrectedRecorded the MathDB record for this result as a registration. No grade moved. source ↗
2026-09-03AddedEntered the registry graded Author verified and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded author verified for verification and ai-led for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). First open case of the big-line-big-clique conjecture. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-19-big-line-big-clique

BibTeX

@misc{whataifound-independent-2026-clique,
  title        = {First open case of the big-line-big-clique conjecture},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-08-19-big-line-big-clique}
}

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