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

A chamber of the vector space F_qⁿ is a set {S₁,…,S_n−1} of subspaces of F_qⁿ where S₁⊂S₂⊂…⊂S_n−1 and dim⁡(S_i)=i for i=1,…,n−1. By Γ_n(q) we denote the graph whose vertices are the chambers of F_qⁿ with two chambers C₁={S₁,…,S_n−1} and C₂={T₁,…,T_n−1} adjacent in Γ_n(q), if S_i∩T_n−i={0} for i=1,…,n−1. The Erdős-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of Γ_n(q). 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 [Formula presented] 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_qⁿ for sufficiently large q, giving an affirmative answer for n even.