  We say that a 0-1 matrix $N$ of size $a\times b$ can be found in a collection
of sets $\mathcal{H}$ if we can find sets $H_{1}, H_{2}, \dots, H_{a}$ in
$\mathcal{H}$ and elements $e_1, e_2, \dots, e_b$ in $\cup_{H \in \mathcal{H}}
H$ such that $N$ is the incidence matrix of the sets $H_{1}, H_{2}, \dots,
H_{a}$ over the elements $e_1, e_2, \dots, e_b$. We prove the following
Ramsey-type result: for every $n\in \N$, there exists a number S(n) such that
in any collection of at least S(n) sets, one can find either the incidence
matrix of a collection of $n$ singletons, or its complementary matrix, or the
incidence matrix of a collection of $n$ sets completely ordered by inclusion.
We give several results of the same extremal set theoretical flavour. For some
of these, we give the exact value of the number of sets required.
