PROVED
This has been solved in the affirmative.
Does there exist $S\subseteq \mathbb{R}^2$ such that every set congruent to $S$ (that is, $S$ after some translation and rotation) contains exactly one point from $\mathbb{Z}^2$?
An old question of Steinhaus. Erdős was 'almost certain that such a set does not exist'.
In fact, such a set does exist, as proved by Jackson and Mauldin
[JaMa02]. Their construction depends on the axiom of choice.
View the LaTeX source
Additional thanks to: Vjekoslav Kovac
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #215, https://www.erdosproblems.com/215, accessed 2026-01-16
