No 5 Points on 2-plane in [N]^d

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 integers $N\geq1$ and $d\geq2$, put $[N]=\{1,\ldots,N\}$ and view $[N]^d$ as a subset of $\mathbb R^d$. Let $\mathcal P_2(\mathbb R^d)$ denote the affine two-dimensional planes, and define \[ M_d(N)=\max\left\{|A|:A\subset[N]^d,\ |A\cap P|\leq4\text{ for every }P\in\mathcal P_2(\mathbb R^d)\right\}. \] The condition includes five collinear points, since a line is contained in an affine two-plane; it does not require the five points to span that plane.

Statement recorded in the supplied source.

Determine the largest possible size $M_d(N)$, and obtain sharp estimates for its growth with $N$ and $d$. [S2]

Short English statement

How large can a subset of a high-dimensional integer grid be if every affine two-plane contains at most four of its points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 135 (GREEN-047). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 74 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.