Maximum Erd{o}s-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension

A chamber of the vector space Fqⁿ is a set ₁,,Sₙ₋₁\ of subspaces of Fqⁿ where S₁⊂ S₂⊂ ⊂ Sₙ₋₁ and (Sᵢ)=i for i=1,,n-1. By Γₙ(q) we denote the graph whose vertices are the chambers of Fqⁿ with two chambers C₁=₁,,Sₙ₋₁\ and C₂=₁,,Tₙ₋₁\ adjacent in Γₙ(q), if Sᵢ∩ Tₙ₋ᵢ=\0\ for i=1,,n-1. The Erd{o}s-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of Γₙ(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 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{o}s-Ko-Rado theorem on chambers of Fqⁿ for sufficiently large q, giving an affirmative answer for n even.