Problem Statement
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 $\mathbb{Z}^2$?
Categories:
Geometry
Progress
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.
Source: erdosproblems.com/215 | Last verified: January 14, 2026