Let X be a finite set of n elements. A family F is a subset of 2 ^X, the power set of X. The generic question is as follows: Supposing that F satisfies certain conditions, determine or estimate the maximum of the size of F. The simplest conditions are:

• no member of F contains another member (Sprener Theorem);

• any two members of F have non-empty intersection (Erdos-Ko-Rado Theorem);

• any two members of F intersect in at least t elements (Katona Theorem).

All the above are by now classical results. Forty years ago the author proposed the more general condition: any r members of F intersect in at least t elements (r and t fixed positive integers). The general answer is still unknown. The lecture will review these and related problems.