Smooth Surfaces Intersecting 2-planes

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 $n\geq3$ and a $C^\infty$ function $f:(-2,2)^{n-2}\to\mathbb R^2$ with $f(0)=0$, define its local graph \[ \Gamma_f=\{(x,f(x)):x\in(-1,1)^{n-2}\}\subset\mathbb R^n. \] An affine two-plane is a translate of a two-dimensional linear subspace of $\mathbb R^n$. Intersection points are distinct points; neither transversality nor finiteness of the full intersection is imposed.

Statement recorded in the supplied source.

Does there exist $n_0$ such that, for every integer $n\geq n_0$ and every such smooth local graph $\Gamma_f$, there is an affine two-plane $P\subset\mathbb R^n$ satisfying \[ |P\cap\Gamma_f|\geq5? \] The question concerns the local geometry of a codimension-two smooth surface, rather than global topology. [S2]

Short English statement

In all sufficiently large dimensions, must every smooth local surface of codimension two meet some affine two-plane in at least five points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 134 (GREEN-046). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 73 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.