OP-00951 Open

Many collinear triples force a cubic curve with many points

Fix $k$ and $\delta > 0$. Let $A \subset \mathbb{R}^2$ be a set of $n$ points with at most $k$ on any line, and suppose that for at least $\delta n^2$ pairs $(x, y) \in A \times A$ the line $xy$ contains a third point of $A$. When $\delta = 1 - O(1/n)$ the conclusion below follows from Green and Tao's work on sets with few ordinary lines, and it is known for sets lying on curves of bounded degree (Elekes and Szabó). Elekes and Szabó conjectured the much weaker statement that a set with $cn^2$ collinear triples has $10$ points on a cubic curve.

Problem. Must some cubic curve pass through at least $cn$ points of $A$, for some $c = c(k, \delta) > 0$?

Source: B. Green, 100 open problems (version of January 2026), Problem 69.

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

Classification

Primary: CO — Combinatorics
Secondary: AG, MG
Keywords: collinear triples, cubic curves, incidence geometry, orchard problem
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{green2013tao,
  author = {Green, Ben and Tao, Terence},
  title = {On sets defining few ordinary lines},
  journal = {Discrete \& Computational Geometry},
  volume = {50},
  number = {2},
  pages = {409--468},
  year = {2013},
  doi = {10.1007/s00454-013-9518-9}
}

@article{elekes2024szabo,
  author = {Elekes, Gy{\"o}rgy and Szab{\'o}, Endre},
  title = {On triple lines and cubic curves: the orchard problem revisited},
  journal = {Discrete \& Computational Geometry},
  volume = {72},
  number = {2},
  pages = {743--763},
  year = {2024},
  doi = {10.1007/s00454-023-00556-3},
  eprint = {1302.5777},
  archivePrefix = {arXiv}
}

@article{barak2013fractional,
  author = {Barak, Boaz and Dvir, Zeev and Wigderson, Avi and Yehudayoff, Amir},
  title = {Fractional {S}ylvester--{G}allai theorems},
  journal = {Proceedings of the National Academy of Sciences of the USA},
  volume = {110},
  number = {48},
  pages = {19213--19219},
  year = {2013},
  doi = {10.1073/pnas.1203737109}
}

@misc{sarkar2026visibility,
  author = {Sarkar, Sohail},
  title = {Visibility cliques, cubic containers, and dense orchard cores},
  year = {2026},
  eprint = {2605.00918},
  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 69; 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-00951, "Many collinear triples force a cubic curve with many points", clopen.solutions.