  A matrix is homogeneous if all of its entries are equal. Let $P$ be a
$2\times 2$ zero-one matrix that is not homogeneous. We prove that if an
$n\times n$ zero-one matrix $A$ does not contain $P$ as a submatrix, then $A$
has an $cn\times cn$ homogeneous submatrix for a suitable constant $c>0$. We
further provide an almost complete characterization of the matrices $P$
(missing only finitely many cases) such that forbidding $P$ in $A$ guarantees
an $n^{1-o(1)}\times n^{1-o(1)}$ homogeneous submatrix. We apply our results to
chordal bipartite graphs, totally balanced matrices, halfplane-arrangements and
string graphs.
