Hall's Marriage Theorem

Also known as
This is called the "Marriage Theorem" for the following reason:

Let $I$ be a set of women.

Suppose that each woman $k$ is romantically interested in a finite set $S_k$ of men.

Suppose also that:
 * each woman would like to marry exactly one of these men

and:
 * each man in $\displaystyle \bigcup_{k \mathop \in I} S_k$ would like to marry at most one woman in $I$.

Then this theorem gives a condition under which it is possible to match every woman to a man.