  Hall's Theorem is a basic result in Combinatorics which states that the
obvious necesssary condition for a finite family of sets to have a transversal
is also sufficient. We present a sufficient (but not necessary) condition on
the sizes of the sets in the family and the sizes of their intersections so
that a transversal exists. Using this, we prove that in a bipartite graph $G$
(bipartition $\{A, B\}$), without 4-cycles, if $\deg(v) \geq \sqrt{2e|A|}$ for
all $v \in A$, then $G$ has a matching of size $|A|$.
