Arbitrarily large integer-distance sets in general position
Anning and Erdős proved that an infinite set of points in the plane with all pairwise distances integers lies on a line. Kreisel and Kurz found seven points in the plane with pairwise integer distances, no three on a line and no four on a circle, answering Erdős's question for seven points. Greenfeld, Iliopoulou and Peluse proved that a set of points in $[-N, N]^2$ with pairwise integer distances, no three on a line and no four on a circle has $O\bigl((\log N)^{O(1)}\bigr)$ points; assuming Lang's conjecture, Ascher, Braune and Turchet proved that the size of such sets is bounded by an absolute constant.
Problem. Is it true that for every positive integer $n$ there is a set $P \subset \mathbb{R}^2$ of $n$ points such that $\|p - q\|_2$ is an integer for all distinct $p, q \in P$, no three points of $P$ lie on a line, and no four points of $P$ lie on a circle? The set may depend on $n$, and no bound on its diameter is prescribed.
Source: Erdős (1975, and later, e.g. 1987, where he calls it an old problem whose origin he cannot trace); see Greenfeld, Iliopoulou and Peluse (2025) and Eppstein (2026), Section 3.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO, NT
- Keywords
- AIM Open Applied Problems, Erdős problems, general position, integer distance sets
References
@incollection{erdos1987,
author = {Erd{\H{o}}s, Paul},
title = {Some combinatorial and metric problems in geometry},
booktitle = {Intuitive Geometry (Si{\'o}fok, 1985)},
series = {Colloq. Math. Soc. J{\'a}nos Bolyai},
volume = {48},
publisher = {North-Holland},
address = {Amsterdam},
year = {1987},
pages = {167--177},
}
@article{eppstein2026anning,
author = {Eppstein, David},
title = {Non-{E}uclidean {E}rd{\H{o}}s--{A}nning theorems},
journal = {J. Comput. Geom.},
volume = {17},
number = {2},
year = {2026},
pages = {46--76},
doi = {10.20382/jocg.v17i2a2},
eprint = {2401.06328},
archivePrefix = {arXiv},
}
@misc{greenfeld2025integer,
author = {Greenfeld, Rachel and Iliopoulou, Marina and Peluse, Sarah},
title = {On integer distance sets},
year = {2025},
eprint = {2401.10821},
archivePrefix = {arXiv},
note = {Version 3},
}
@article{anning1945,
author = {Anning, Norman H. and Erd{\H{o}}s, Paul},
title = {Integral distances},
journal = {Bull. Amer. Math. Soc.},
volume = {51},
number = {8},
year = {1945},
pages = {598--600},
doi = {10.1090/S0002-9904-1945-08407-9},
}
@article{kreisel2008,
author = {Kreisel, Tobias and Kurz, Sascha},
title = {There are integral heptagons, no three points on a line, no four on a circle},
journal = {Discrete Comput. Geom.},
volume = {39},
number = {4},
year = {2008},
pages = {786--790},
doi = {10.1007/s00454-007-9038-6},
}
@article{ascher2020,
author = {Ascher, Kenneth and Braune, Lucas and Turchet, Amos},
title = {The {E}rd{\H{o}}s--{U}lam problem, {L}ang's conjecture and uniformity},
journal = {Bull. Lond. Math. Soc.},
volume = {52},
number = {6},
year = {2020},
pages = {1053--1063},
doi = {10.1112/blms.12381},
}
@misc{aim-integer-distance-general-position,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Arbitrarily large integer-distance sets in general position}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/558-integer-distance-general-position.md},
note = {Entry 558 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00870, "Arbitrarily large integer-distance sets in general position", clopen.solutions.