  Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G
on vertex sets X,Y that satisfies the following condition; one can add the
edges between X and Y that do not belong to G one after the other so that
whenever a new edge is added, a new copy of K_{p,q} is created. The problem of
bounding W_n(p,q), and its natural hypergraph generalization, was introduced by
Balogh, Bollob\'as, Morris and Riordan. Their main result, specialized to
graphs, used algebraic methods to determine W_n(1,q).
  Our main results in this paper give exact bounds for W_n(p,q), its hypergraph
analogue, as well as for a new variant of Bollob\'as's Two Families theorem. In
particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n
then
  W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2.
  Our proof applies a reduction to a multi-partite version of the Two Families
theorem obtained by Alon. While the reduction is combinatorial, the main idea
behind it is algebraic.
