2-bases of quadruples

Let Beta(n, <= 4) denote the subsets of [n] := {1, 2,..., n} of at most 4 elements. Suppose that F is a set system with the property that every member of B can be written as a union of (at most) two members of F. (Such an F is called a 2-base of B.) Here we answer a question of Erdos proving that [GRAPHICS] and this bound is best possible for n >= 8.