A Simple Condition for the Existence of Transversals

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) ≥ √ 2e|A| for all v ∈ A, then G has a matching of size |A|.