Collinear 4-tuples Force Collinear 5-tuples

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 $A\subset\mathbb R^2$, define \[ t_4(A)=\left|\{B\subset A:|B|=4\text{ and all points of }B\text{ lie on one line}\}\right|. \] Thus $t_4$ counts unordered subsets of four distinct points. If no five points are collinear, each four-point subset corresponds to a line meeting $A$ in exactly four points; this matches the line-counting convention of the cited source.

Statement recorded in the supplied source.

For every real $c>0$, is there an integer $n_0(c)$ such that every finite $A\subset\mathbb R^2$ with $|A|=n\geq n_0(c)$ and $t_4(A)\geq c n^2$ contains five distinct collinear points? Equivalently, if \[ T_4(n)=\max\{t_4(A):A\subset\mathbb R^2,\ |A|=n,\text{ no five points of }A\text{ are collinear}\}, \] is $T_4(n)/n^2\longrightarrow0$ as $n\to\infty$? [S2]

Short English statement

Must every large point set with a fixed positive quadratic number of collinear four-point subsets contain five collinear points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 132 (GREEN-044). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 József Solymosi and Miloš Stojaković, Many collinear $k$-tuples with no $k+1$ collinear points, Discrete & Computational Geometry 50 (2013), 811--820, §1, Erdős conjecture for $r=5,k=4$. https://arxiv.org/abs/1107.0327.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 71 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.