I'm a little late in realizing it, but today is Pigeon-hole Day. Festivities include thinking about awesome applications of the Pigeon-hole Principle. So let's come up with some. As always with these kinds of questions, please only post one answer per post so that it's easy for people to vote on them.

Allow me to start with an example:

Brouwer's fixed point theorem can be proved with the Pigeon-hole Principle via Sperner's lemma. There's a proof in Proofs from The Book (unfortunately, the google books preview is missing page 148)

By the way, if you happen to be in Evans at Berkeley today, come play musical chairs at tea!

I'm a little late in realizing it, but today is Pigeon-hole Day. Festivities include thinking about awesome applications of the Pigeon-hole Principle. So let's come up with some. As always with these kinds of questions, please only post one answer per post so that it's easy for people to vote on them.

Allow me to start with an example:

Brouwer's fixed point theorem can be proved with the Pigeon-hole Principle via Sperner's lemma. There's a proof in Proofs from The Book (unfortunately, the google books preview is missing page 148)