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.

Examples

Suppose $\$x$ is allocated to healthcare and some treatment costs $\$y$ then there are $\$x/\$y$ times that the treatment can be administered. After that, everyone else goes without treatment. If insurance denies no one, then that means that the first $\$x/\$y$ people get the treatment and no one else does.