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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$\{|A|:A\in f\}$ is called the type of $f$. Two flags $f$ and $f'$ are ingeneral position (with respect to $S$) when $A\cap B=\emptyset$ or $A\cup B=S$for all $A\in f$ and $B\in f'$. We study sets of flags of a fixed type $T$ thatare mutually not in general position and are interested in the largestcardinality of these sets. This is a generalization of the classicalErd\"os-Ko-Rado problem. We will give some basic facts and determine thelargest cardinality in several non-trivial cases. For this we will definegraphs whose vertices are flags and the problem is to determine theindependence number of these graphs.

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