  We show tight necessary and sufficient conditions on the sizes of small
bipartite graphs whose union is a larger bipartite graph that has no large
bipartite independent set. Our main result is a common generalization of two
classical results in graph theory: the theorem of K\H{o}v\'{a}ri, S\'{o}s and
Tur\'{a}n on the minimum number of edges in a bipartite graph that has no large
independent set, and the theorem of Hansel (also Katona and Szemer\'{e}di,
Krichevskii) on the sum of the sizes of bipartite graphs that can be used to
construct a graph (non-necessarily bipartite) that has no large independent
set.
  As an application of our results, we show how they unify the underlying
combinatorial principles developed in the proof of tight lower bounds for
depth-two superconcentrators.
