Anti Pigeonhole Principle

The Anti Pigeonhole Principle is that if there are $n\in \mathbb {N}$ pigeons and $m\in \mathbb {M}$ pigeonholes such that $n<m$ then there must be some pigeonholes that are empty.

For sets $A$ and $B$ and function $f:A\to B$ with $|A|=n;|B|=m$ where $n<m$, then $\exists b\in B$ such that $|f^{-1}({b})|=0$.

Also, see Pigeonhole principle.