IrMO 2018 Problem 4

We say that a rectangle with side lengths $a$ and $b$ fits inside a rectangle with side lengths $c$ and $d$ if either ( $a \le c$ and $b \le d$ ) or ( $a \le d$ and $b \le c$ ). For instance, a rectangle with side lengths $1$ and $5$ fits inside another rectangle with side lengths $1$ and $5$ , and also fits inside a rectangle with side lengths $6$ and $2$ . Suppose $S$ is a set of $2019$ rectangles, all with integer side lengths between $1$ and $2018$ inclusive. Show that there are three rectangles $A$ , $B$ , and $C$ in $S$ such that $A$ fits inside $B$ , and $B$ fits inside $C$ .