OP-00990 Open

Large subsets of the grid with no five points on a plane

Fix $d$. Green does not know whether a subset of $[N]^d$ with no five points on a $2$-dimensional plane can have size $N^{d - 100}$. Since conics are planar and any five points of a plane lie on a conic, asking for no five points on a conic is a nearly equivalent formulation.

Problem. How large can a subset of $[N]^d$ be if no $5$ of its points lie on a common $2$-plane?

Source: B. Green, 100 open problems (version of January 2026), Problem 74.

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CO — Combinatorics
Secondary: MG
Keywords: conics, discrete geometry, general position, lattice points
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{bays2021breuillard,
  author = {Bays, Martin and Breuillard, Emmanuel},
  title = {Projective geometries arising from {E}lekes--{S}zab{\'o} problems},
  journal = {Annales Scientifiques de l'{\'E}cole Normale Sup{\'e}rieure (4)},
  volume = {54},
  number = {3},
  pages = {627--681},
  year = {2021},
  doi = {10.24033/asens.2467}
}

@article{pach2003midpoints,
  author = {Pach, J{\'a}nos},
  title = {Midpoints of segments induced by a point set},
  journal = {Geombinatorics},
  volume = {13},
  number = {2},
  pages = {98--105},
  year = {2003}
}

@inproceedings{dvir2012lovett,
  author = {Dvir, Zeev and Lovett, Shachar},
  title = {Subspace evasive sets},
  booktitle = {Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC '12)},
  pages = {351--358},
  year = {2012},
  doi = {10.1145/2213977.2214010}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 74; version of January 2026, most recent update December 2025},
  year = {2026}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00990, "Large subsets of the grid with no five points on a plane", clopen.solutions.