Extremal problems for finite sets

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 (Erdős-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.