Question 11
$S$ is a horizontal strip in the plane. $n$ lines are drawn so that no three are collinear and every pair intersects within the strip. A path starts at the bottom border of the strip and consists of a sequence of segments from the $n$ lines. The path must change line at each intersection and must always move upwards. Show that:
(1) there are at least $n/2$ disjoint paths;
(2) there is a path of at least $n$ segments;
(3) there is a path involving not more than $n/2 + 1$ of the lines; and
(4) there is a path that involves segments from all $n$ lines.
(1) there are at least $n/2$ disjoint paths;
(2) there is a path of at least $n$ segments;
(3) there is a path involving not more than $n/2 + 1$ of the lines; and
(4) there is a path that involves segments from all $n$ lines.