Integer distances in general position

How many points can one place in the plane, no three on a line and no four on a circle, so that all pairwise distances are integers? Sets of seven such points have been known since 2008, but nobody has found eight. Erdős asked whether such sets exist for every size.

Problem

Call a finite set $S\subset\mathbb{R}^2$ an integral point set in general position if no three points of $S$ lie on a line, no four lie on a circle, and $\lvert x-y\rvert\in\mathbb{Z}$ for all $x,y\in S$. Is it true that for every $n\ge4$ there is such a set with $n$ points?

Background

Anning and Erdős proved in 1945 that an infinite set of points in the plane with all pairwise distances integral must lie on a line, so integral point sets that are not collinear are finite. The question here asks whether, under the stronger requirement of general position, they can still be arbitrarily large. Harborth found an example with five points, and Kreisel and Kurz (2008) found integral heptagons in general position. The problem is closely related to the Erdős–Ulam problem on dense rational distance sets.

What is known

• Constructions. Five points (Harborth) and seven points (Kreisel and Kurz, 2008); no example with eight points is known.
• Conditional bound. Ascher, Braune and Turchet (2020) showed that, assuming Lang's conjecture, the size of such sets is uniformly bounded, which would answer the question negatively.
• Sparsity. Greenfeld, Iliopoulou and Peluse (2024) proved unconditionally that an integral point set in general position inside $[-N,N]^2$ has at most $(\log N)^{O(1)}$ points.

What would count as progress

• Eight points in general position with all distances integral; given exact coordinates (for example in a number field), the conditions can be verified symbolically.
• Searches that exclude eight-point examples with all coordinates or distances below a stated bound, with code and logs.
• An unconditional uniform bound on the size of such sets, or an improvement of the polylogarithmic sparsity bound.
• A formal proof against the Lean statement in formal-conjectures.

References

1. N. H. Anning and P. Erdős, Integral distances, Bull. Amer. Math. Soc. 51 (1945), 598–600. doi:10.1090/S0002-9904-1945-08407-9
2. T. Kreisel and S. Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), 786–790. doi:10.1007/s00454-007-9038-6
3. K. Ascher, L. Braune and A. Turchet, The Erdős–Ulam problem, Lang's conjecture, and uniformity (2020). arXiv:1901.02616
4. R. Greenfeld, M. Iliopoulou and S. Peluse, On integer distance sets (2024). arXiv:2401.10821

Source

Imported from Erdős Problems 213, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.