A question of Erd\H{o}s and T\'{o}th. The answer to the first question is yes (which Erd\H{o}s gave Danzer \$10 for). There is no prize mentioned in \cite{Er87b} for the (still open) second question.

There are two examples Erd\H{o}s gives in \cite{Er87b}, the {IMAGE=1071-one,first} by Danzer, the {IMAGE=1071-two,second} by an unnamed participant.

In \cite{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.


References


[Er87b] Erd\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\'{o}fok, 1985) (1987), 167-177.