Erdős Problem #1071

- $10

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?

#1071: [Er87b,p.173]

geometry

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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.

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This page was last edited 16 January 2026.

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Formalised statement? Yes

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Additional thanks to: Boris Alexeev and Wouter van Doorn

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