Ramsey-type results on singletons, co-singletons and monotone sequences in large collections of sets

We say that a 0–1 matrix N of size a×b can be found in a collection of sets H if we can find sets H1,H2,…,Ha in H and elements e1,e2,…,eb in ∪H∈HH such that N is the incidence matrix of the sets H1,H2,…,Ha over the elements e1,e2,…,eb. We prove the following Ramsey-type result: for every n∈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.