Anti Pigeonhole Principle

The Anti Pigeonhole Principle is that if there are $n\in \mathbb{\mathbb{N}}$ 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 $\mathrm{\exists }b\in B$ such that $|f^{-1}(b)|=0$.

Also, see Pigeonhole principle.

Examples

Suppose $\mathrm{\$}x$ is allocated to healthcare and some treatment costs $\mathrm{\$}y$ then there are $\mathrm{\$}x/\mathrm{\$}y$ 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 $\mathrm{\$}x/\mathrm{\$}y$ people get the treatment and no one else does.