Maximum Erdős-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension

Abstract

A chamber of the vector space ${\mathbb{F}}_q^n$ is a set $\{S_1,\dots ,S_{n-1}\}$ of subspaces of ${\mathbb{F}}_q^n$ where $S_1\subset S_2\subset \dots \subset S_{n-1}$ and $\dim (S_i)=i$ for $i=1,\dots ,n-1$. By ${\Gamma }_n(q)$ we denote the graph whose vertices are the chambers of ${\mathbb{F}}_q^n$ with two chambers $C_1=\{S_1,\dots ,S_{n-1}\}$ and $C_2=\{T_1,\dots ,T_{n-1}\}$ adjacent in ${\Gamma }_n(q)$, if $S_i\cap T_{n-i}=\{0\}$ for $i=1,\dots ,n-1$. The Erdős-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of ${\Gamma }_n(q)$. The independence number of this graph was determined in [7] for $n$ even and given a subspace $P$ of dimension one, the set of all chambers whose subspaces of dimension $\frac{n}{2}$ contain $P$ attains the bound. The dual example of course also attains the bound. It remained open in [7] whether or not these are all maximum independent sets. Using a description from [6] of the eigenspace for the smallest eigenvalue of this graph, we prove an Erdős-Ko-Rado theorem on chambers of ${\mathbb{F}}_q^n$ for sufficiently large $q$, giving an affirmative answer for n even.