The Pigeonhole Principle

Abstract

The pigeonhole principle (also known as Dirichlet’s principle) states the “obvious” fact that n+1 pigeons cannot sit in n holes so that every pigeon is alone in its hole. More generally, the pigeonhole principle states the following:

If a set consisting of at least rs+1 objects is partitioned into r classes, then some class receives at least s+1 objects.

Its truth is easy to verify: if every class receives at most s objects, then a total of at most rs objects have been distributed. To see that the result is best possible, observe that a set with at most rs points can be divided into r groups with at most s points in each group; hence none of the groups contains s+1 points.