Generalised Empty Hexagon Conjecture

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 $P\subset\mathbb R^2$, an empty hexagon consists of six points $S\subseteq P$ in strictly convex position such that \[\operatorname{conv}(S)\cap P=S.\] Thus no additional point lies either inside or on the boundary of the hexagon.

Statement recorded in the supplied source.

For every integer $\ell\geq3$, does there exist an integer $f(\ell)$ such that every finite $P\subset\mathbb R^2$ with $|P|\geq f(\ell)$ contains $\ell$ collinear points or an empty hexagon? [S2]

Short English statement

If collinear subsets have a fixed bounded size, must every sufficiently large planar point set contain a convex hexagon with no other point in its closed hull?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3091 (OPG-59923). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wood, David R., Generalised Empty Hexagon Conjecture, Open Problem Garden, node 59923. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/generalised_empty_hexagon_conjecture.