Big-Line-Big-Clique Conjecture for planar point sets

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The Big-Line-Big-Clique Conjecture for planar point sets

   Let $P$ be a finite set of points in the plane. Two points are visible if the open segment between them contains no point of $P$, and the visibility graph $\mathcal{V}(P)$ has vertex set $P$ with visible pairs as edges. Big-Line-Big-Clique Conjecture. For all positive integers $k$ and $\ell$ there is an integer $n$ such that every finite set $P$ of at least $n$ points in the plane contains $\ell$ collinear points or $k$ pairwise visible points, that is, a $k$-clique in $\mathcal{V}(P)$. The conjecture is true for $k\leq5$ or $\ell\leq3$, but remains open for $k=6$ or $\ell=4$.

   source: Attila Pór and David R. Wood, “On Visibility and Blockers”, arXiv:0912.1150 (2009).