Five points of a codimension-two surface on a two-plane
The question is local: one may take $\Gamma = \{(x, f(x)) : x \in (-1, 1)^{n-2}\} \subset \mathbb{R}^n$ for a smooth function $f : (-2, 2)^{n-2} \to \mathbb{R}^2$ with $f(0) = 0$. More generally, for a codimension-$d$ surface one asks that some $d$-dimensional plane meet it in $f(n, d)$ points. A convex hypersurface shows that $f(n, 1) = 2$, and $f(n, d) \ge 2^d$ for large $n$, by the argument that finds Hilbert cubes in large sets in abelian groups. That $f(n, d)$ is finite was sketched by Thom and worked out by Chaperon and Meyer.
Problem. For $n$ sufficiently large, must every smooth codimension-$2$ surface $\Gamma \subset \mathbb{R}^n$ contain $5$ points lying in a common $2$-dimensional plane?
Source: B. Green, 100 open problems (version of January 2026), Problem 73.
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO, DG
- Keywords
- finite geometry of submanifolds, Hilbert cubes, plane sections, Thom's theorem
References
@incollection{thom1969ordre,
author = {Thom, Ren{\'e}},
title = {Sur les vari{\'e}t{\'e}s d'ordre fini},
booktitle = {Global Analysis (Papers in Honor of K. Kodaira)},
pages = {397--401},
publisher = {University of Tokyo Press},
year = {1969}
}
@article{chaperon2009thom,
author = {Chaperon, Marc and Meyer, Daniel},
title = {On a theorem of {R}en{\'e} {T}hom in ``g{\'e}om{\'e}trie finie''},
journal = {L'Enseignement Math{\'e}matique (2)},
volume = {55},
number = {3-4},
pages = {329--357},
year = {2009},
doi = {10.4171/LEM/55-3-6}
}
@article{makeev1990knaster,
author = {Makeev, V. V.},
title = {The {K}naster problem and almost spherical sections},
journal = {Mathematics of the USSR-Sbornik},
volume = {66},
number = {2},
pages = {431--438},
year = {1990},
doi = {10.1070/SM1990v066n02ABEH001179}
}
@misc{sarkar2026polygons,
author = {Sarkar, Sohail},
title = {Polygons and coplanar points on submanifolds of codimension 2},
year = {2026},
eprint = {2610.14032},
archivePrefix = {arXiv}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 73; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in $\mathbb{R}^n$ must meet some affine two-plane in five points when $n$ is large. In this paper, we answer this question for every $n \geq 6$. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.14032v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01042, "Five points of a codimension-two surface on a two-plane", clopen.solutions.