TITLE: Show that if there are 101 people of different heights standing in a line
QUESTION [5 upvotes]: Show that if there are 101 people of different heights
standing in a line, it is possible to ﬁnd 11 people in the
order they are standing in the line with heights that are
either increasing or decreasing.

REPLY [3 votes]: Define a partial order on the set of people by stipulating that $x\lt y$ means that person $x$ is shorter than $y$ and is standing behind $y$. The question asks us to show that this partially ordered set of size $101$ contains either a chain or an antichain of size $11$. The simpler way to do this is to observe that a partially ordered set, which contains no $11$-element chain, is the union of $10$ antichains, one of which must contain more than $10$ elements by the pigeonhole principle. The fancier alternative is to quote Dilworth's chain decomposition theorem: if there is no $11$-element antichain, then the poset is the union of $10$ chains, etc.