Extremal set theory

One of the simplest objects are finite sets. For a positive integer n let [n] denote the set of integers 1,2,...,n. Let 2^[n] be the power set of [n] (it consists of 2^n subsets. Any subset F of 2^[n] is called a family. Extremal set theory is dealing with (best possible) upper bounds on |F|, the number of subsets in F, subject to some conditions imposed on F. For example, the classical theorem of Sperner (1928) states that if no member of F contains another then |F| is at most the largest binomial coefficient. We are going to present several results and problems of similar flavour.