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
Independent commentary
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. 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.
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.
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Entry history (1 event)
- AddedEntered 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
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-cliqueBibTeX
@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}
}