Erdős-Szekeres with Visibility

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite set $A\subset\mathbb R^2$, two distinct points $x,y\in A$ are visible relative to $A$ if the open line segment between them contains no point of $A$; equivalently, $[x,y]\cap A=\{x,y\}$. A subset is pairwise visible if each pair of its distinct members is visible relative to the entire ambient set $A$, including points outside the subset.

Statement recorded in the supplied source.

For every pair of integers $k,\ell\geq2$, does there exist an integer $n_0(k,\ell)$ such that every finite set $A\subset\mathbb R^2$ with $|A|\geq n_0(k,\ell)$ contains either $k$ distinct collinear points or $\ell$ pairwise visible points? [S2]

Short English statement

Does every sufficiently large planar point set contain either many points on one line or a prescribed number of points with no other point blocking any connecting segment?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 131 (GREEN-043). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Zachary Abel, Brad Ballinger, Prosenjit Bose, Sébastien Collette, Vida Dujmović, Ferran Hurtado, Scott D. Kominers, Stefan Langerman, Attila Pór and David R. Wood, Every Large Point Set Contains Many Collinear Points or an Empty Pentagon, Graphs and Combinatorics 27 (2011), 47--60, §1.4, Conjecture 4. https://arxiv.org/abs/0904.0262.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 70 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.