Problem Statement
Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maximal with respect to this property?
Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?
Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?
Categories:
Geometry
Progress
A question of Erdős and Tóth. The answer to the first question is yes (which Erdős gave Danzer \$10 for). There is no prize mentioned in [Er87b] for the (still open) second question.There are two examples Erdős gives in [Er87b], the first by Danzer, the second by an unnamed participant.
In [Er87b] he further asks what happens if the unit line segments are rotated/translated copies of $[0,1]$ that are allowed to intersect only at their endpoints.
Source: erdosproblems.com/1071 | Last verified: January 19, 2026