  For a bipartite graph G, let h(G) be the largest t such that either G or the
bipartite complement of G contain K_{t,t}. For a class F of graphs, let h(F)=
min {h(G): G\in F}. We say that a bipartite graph H is strongly acyclic if
neither H nor its bipartite complement contain a cycle. By Forb(n, H) we denote
a set of bipartite graphs with parts of sizes n each, that do not contain H as
an induced bipartite subgraph respecting the sides. One can easily show that
h(Forb(n,H))= O(n^{1-s}) for a positive s if H is not strongly acyclic. Here,
we prove that h(Forb(n, H)) is linear in n for all strongly acyclic graphs
except for four graphs.
