Erdős-Ko-Rado sets of flags of finite sets

Abstract

A flag of a finite set S is a set f of non-empty proper subsets of S such that A⊆B or B⊆A for all A,B∈f. The set {|A|:A∈f} is called the type of f. Two flags f and f′ are in general position (with respect to S) when A∩B=∅ or A∪B=S for all A∈f and B∈f′. 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 Erdős-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.