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

TL;DR

This paper generalizes the Erdös-Ko-Rado problem to flags of finite sets, analyzing the maximum size of collections of such flags that are pairwise not in general position, using graph-theoretic methods.

Contribution

It introduces a new framework for Erdös-Ko-Rado type problems involving flags of finite sets and determines maximum sizes in several cases.

Findings

Established basic properties of flag sets in this context

Determined maximum cardinalities for specific non-trivial cases

Connected the problem to independence numbers of certain graphs

Abstract

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 in general 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 $T$ 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.