1. Introduction. Let X be an n-element set and F C 2 X a family of distinct subsets of X. Suppose that the members of F satisfy some conditions. What is the maximum (or minimum) value of |F|—this is the generic problem in extremal set theory. There have been far too many papers and results in this area to be overviewed in such a short paper. Therefore, we will only deal with some intersection theorems. The simplest is