Many collinear four-point sets force five points on a line
A collinear $4$-tuple of a set $A \subset \mathbb{R}^2$ is a set of four points of $A$ on a line. A construction of Solymosi and Stojaković gives sets of $n$ points with $e^{-c\sqrt{\log n}} n^2$ collinear $4$-tuples and no five points on a line. Much more than the statement below should be true, for instance $100$ points on a line.
Problem. Let $A \subset \mathbb{R}^2$ be an $n$-point set having $cn^2$ collinear $4$-tuples. Must some line contain $5$ points of $A$?
Source: P. Erdős, asked on numerous occasions; Erdős problem #101. Listed as Problem 71 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MG
- Keywords
- collinear points, Erdős problems, incidence geometry, rich lines
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{solymosi2013stojakovic,
author = {Solymosi, J{\'o}zsef and Stojakovi{\'c}, Milo{\v s}},
title = {Many collinear $k$-tuples with no $k+1$ collinear points},
journal = {Discrete \& Computational Geometry},
volume = {50},
number = {3},
pages = {811--820},
year = {2013},
doi = {10.1007/s00454-013-9526-9}
}
@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}
}
@misc{erdosproblems101,
author = {Bloom, Thomas F.},
title = {Erd{\H o}s Problem \#101},
howpublished = {\url{https://www.erdosproblems.com/101}},
note = {Accessed 2026-10-26}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 71; 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 It will not change if the problem is solved, reclassified or withdrawn.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-00952, "Many collinear four-point sets force five points on a line", clopen.solutions.