No four subsets forming an N

We survey results concerning the maximum size of a family F of subsets of an n -element set such that a certain configuration is avoided. When F avoids a chain of size two, this is just Sperner's theorem. Here we give bounds on how large Y can be such that no four distinct sets A, B, C, D is an element of F, satisfy A subset of B, C subset of B, C subset of D. In this case, the maximum size satisfies [GRAPHICS] which is very similar to the best-known bounds for the more restrictive problem of F avoiding three sets B, C, D such that C subset of B, C subset of D. (C) 2007 Elsevier Inc. All rights reserved.