Erd\"os-Ko-Rado sets of flags of finite sets

A flag of a finite set $S$ is a set $f$ of non-empty proper subsets of $S$ such that $A\subseteq B$ or $B\subseteq A$ for all $A,B\in f$. The set $\{\mathrm{\mid }A\mathrm{\mid }:A\in f\}$ is called the type of $f$. Two flags $f$ and $f^{\mathrm{\prime }}$ are in general position (with respect to $S$) when $A\cap B=\mathrm{\varnothing }$ or $A\cup B=S$ for all $A\in f$ and $B\in f^{\mathrm{\prime }}$. We study sets of flags of a fixed type $T$ that are mutually not in general position and are interested in the largest cardinality of these sets. This is a generalization of the classical Erd"os-Ko-Rado problem. We will give some basic facts and determine the largest cardinality in several non-trivial cases. For this we will define graphs whose vertices are flags and the problem is to determine the independence number of these graphs.