big-line-big-clique conjecture

About 21 years old · traced to 2005

For a finite set $S$ of points in the Euclidean plane, say that two distinct points $p,q\in S$ are mutually visible (with respect to $S$ ) if no point of $S\setminus\{p,q\}$ lies on the open line segment with endpoints $p$ and $q$ . A subset $T\subseteq S$ is a clique if every two distinct points of $T$ are mutually visible with respect to $S$ .

For all positive integers $k$ and $\ell$ there exists a positive integer $n_{k,\ell}$ such that every set $S$ of at least $n_{k,\ell}$ points in the plane contains $\ell$ collinear points, or a clique of $k$ points, or both.

References

Primary source

Wikipedia

Additional references

Wikipedia, Big-line-big-clique conjecture, the article this problem comes from.

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims to settle the first unresolved finite case, but its result has not yet been independently verified.

The conjecture, introduced by Kára, Pór, and Wood, predicts that sufficiently large finite planar point sets contain either many collinear points or a large mutually visible clique. The general statement remains broader than the newly claimed case.

Known results

The conjecture is proved when $k\leq 5$ or $\ell\leq 3$ .
Kára, Pór, and Wood proved the case $k=4$ for all $\ell$ .
Abel et al. proved the case $k=5$ for all $\ell$ .
The finite cases $k=6$ or $\ell=4$ were the principal unresolved cases before 2026.

August 2026 claimed resolution

Bonnet’s preprint claims an explicit finite threshold forcing $4$ collinear points or a $6$ -clique, thereby resolving the $(k,\ell)=(6,4)$ case. It is not yet peer reviewed or independently verified. A separate 2026 preprint gives substantial structural partial results but explicitly leaves the full conjecture unresolved.

Current status (as of August 2026): The $(6,4)$ case is claimed solved by a new preprint, but remains unverified; the full conjecture and other parameter cases remain open.

Sources

No solutions have been posted yet.