Cocliques in the Kneser graph on (n − 1,n)-flags of PG(2n,q)

In the finite projective space PG$(2n,q)$ we consider flags of type $(n-1,n)$, that is, pairs $(A,B)$ consisting of an $(n-1)$-space A and an n-space B that are incident. Two such flags (A₁,B₁) and (A₂,B₂) are opposite if A₁∩ B₂=A₂∩ B₁=∅. Let Γ₂ₙ be the graph whose vertices are the flags of type $(n-1,n)$ of PG$(2n,q)$, with two vertices being adjacent if the corresponding flags are opposite. Using the Erdős-Matching theorem for vector spaces shown by Ihringer, we determine, for q large enough, the largest cocliques of Γ₂ₙ and obtain a stability result. This EKR-type theorem proves a conjecture of D'haeseleer, Metsch and Werner.