On the Erd\H{o}s-Ko-Rado problem of flags with type $\{1, n-3 \}$ of finite sets

Abstract

A flag of a finite set $S$ is a set $f$ of non-empty, proper subsets of $S$, such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. Two flags $f_1$ and $f_2$ of $S$ are opposite if $X_1\cap X_2=\emptyset$, or $X_1\cup X_2=S$ for all $X_1\in f_1$ and $X_2\in f_2$. The set $\{|X| \mid X\in f \}$ is the type of a flag $f$. A set of pairwise non-opposite flags is an Erd\H{o}s-Ko-Rado set. In 2022 Metsch posed the problem of determining the maximum size of all Erd\H{o}s-Ko-Rado sets of flags of type $T$ with $|T|=2$. We contribute towards this by determining the maximum size for flags of type $\{ 1,n-3\}$ for finite sets with $n$ elements. Furthermore we answer an open questions of Metsch regarding a small case.