  A family of sets F is said to be union-closed if A \cup B is in F for every A
and B in F. Frankl's conjecture states that given any finite union-closed
family of sets, not all empty, there exists an element contained in at least
half of the sets. Here we prove that the conjecture holds for families
containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight
4-subsets of a 6-set, extending work of Poonen and Vaughan. As an application
we prove the conjecture in the case that the largest set has at most nine
elements, extending a result of Gao and Yu. We also pose several open
questions.
