  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 $S_1\subset
S_2\subset \dotso \subset S_{n-1}$ 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 Erd\H{o}s-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 $\frac n2$ 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
Erd\H{o}s-Ko-Rado theorem on chambers of $\mathbb{F}_q^n$ for sufficiently
large $q$, giving an affirmative answer for n even.
