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

A chamber of the vector space $F_q^n$ is a set $\{S_1,\dots ,S_{n-1}\}$ of subspaces of $F_q^n$ where $S_1\subset S_2\subset \dots \subset S_{n-1}$ and $\dim \left(S_i\right)=i$ for $i=1,\dots ,n-1$. By ${\Gamma }_n\left(q\right)$ we denote the graph whose vertices are the chambers of $F_q^n$ with two chambers $C_1=\{S_1,\dots ,S_{n-1}\}$ and $C_2=\{T_1,\dots ,T_{n-1}\}$ adjacent in ${\Gamma }_n\left(q\right)$, if $S_i\cap T_{n-i}=\left\{0\right\}$ for $i=1,\dots ,n-1$. The Erdős-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of ${\Gamma }_n\left(q\right)$. The independence number of this graph was determined in [5] 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 [5] 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 $F_q^n$ for sufficiently large q, giving an affirmative answer for n even.