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.