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 $(A_1,B_1)$ and $(A_2,B_2)$ are opposite if $A_1\cap B_2=A_2\cap B_1=\emptyset$. Let $Γ_{2n}$ 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 $Γ_{2n}$ and obtain a stability result. This EKR-type theorem proves a conjecture of D’haeseleer, Metsch and Werner.

💡 Research Summary

The paper investigates the Kneser‑type graph Γ₂ₙ whose vertices are (n‑1,n) flags in the finite projective space PG(2n,q). A flag is a pair (A,B) with an (n‑1)-dimensional subspace A contained in an n‑dimensional subspace B. Two flags (A₁,B₁) and (A₂,B₂) are declared opposite when A₁∩B₂=∅ and A₂∩B₁=∅; opposite flags are adjacent in Γ₂ₙ. The central problem is to determine the size and structure of the largest cocliques (sets of pairwise non‑adjacent vertices) of this graph for large q.

The authors first present two natural families of cocliques (Example 1.1). In the first family a hyperplane H of PG(2n,q) is fixed together with either a point X∈H or a (2n‑2)-subspace X⊂H; the coclique consists of all flags (A,B) such that either B⊆H or A is incident with both X and H. Dually, the second family fixes a point P and either a hyperplane through P or a line through P, and takes all flags with P⊆A or B incident with both X and P. Both families have the same cardinality \