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 F can be such that no four distinct sets A,B,C,D∈F satisfy A⊂B, C⊂B, C⊂D. In this case, the maximum size satisfies , 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⊂B, C⊂D.