Collinear Triples and Cubic Curves

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.

An affine line through two distinct points $x,y\in\mathbb R^2$ is denoted by $\ell(x,y)$. For a finite set $A\subset\mathbb R^2$, let \[ T(A)=\left|\{(x,y)\in A^2:x\ne y,\ (\ell(x,y)\cap A)\setminus\{x,y\}\ne\varnothing\}\right|. \] Pairs are ordered. A cubic curve means the zero set of a nonzero real polynomial in two variables of total degree at most three; it may be reducible.

Statement recorded in the supplied source.

For every integer $k\geq2$ and every real $\delta>0$, does there exist $c=c(k,\delta)>0$ such that, for every finite set $A\subset\mathbb R^2$ of cardinality $n\geq1$ with at most $k$ points on any line and $T(A)\geq\delta n^2$, there is a nonzero $F\in\mathbb R[X,Y]$ of total degree at most three with \[ |\{a\in A:F(a)=0\}|\geq cn? \] [S2]

Short English statement

Must a planar point set with bounded line multiplicities and many pairs lying in collinear triples have a positive proportion of its points on one cubic curve?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 130 (GREEN-042). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 69 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.