General position subsets

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.

A finite subset of $\mathbb R^2$ is in general position if it contains no three collinear points. For $n\geq3$ define \[f(n)=\min\left\{N\in\mathbb N:\ \forall P\subset\mathbb R^2\text{ finite},\ |P|\geq N\Longrightarrow\left(\exists\text{ a line containing }n\text{ points of }P\right)\ \text{or}\ \left(\exists S\subseteq P,\ |S|=n,\ S\text{ in general position}\right)\right\}.\]

Statement recorded in the supplied source.

Determine $f(n)$, including its asymptotic growth. In particular, is there an absolute $C$ such that $f(n)\leq Cn^2$ for all $n\geq3$? [S2]

Short English statement

How many planar points force either n collinear points or n points with no three collinear, and is a quadratic number always enough?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3090 (OPG-59914). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gowers, Timothy, General position subsets, Open Problem Garden, node 59914. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/general_position_subsets.