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 x b can be found in a collection of sets H if we can find sets H-1, H-2,..., H-a in H and elements e(1), e(2),..., e(b) in boolean ORHis an element ofHH such that N is the incidence matrix of the sets H-1, H-2,..., H-a over the elements e(1), e(2),..., e(b). We prove the following Ramsey-type result: for every n is an element of 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. (C) 2003 Elsevier Ltd. All rights reserved.