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

Philipp Heering

Abstract

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.

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

Abstract

In the finite projective space PG we consider flags of type , that is, pairs consisting of an -space and an -space that are incident. Two such flags and are opposite if . Let be the graph whose vertices are the flags of type of PG, 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 large enough, the largest cocliques of and obtain a stability result. This EKR-type theorem proves a conjecture of D'haeseleer, Metsch and Werner.
Paper Structure (4 sections, 11 theorems, 7 equations)

This paper contains 4 sections, 11 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Two colors
  3. Many red subspaces
  4. Few red subspaces

Key Result

Theorem 1.2

For $n\geq 3$ let $\Gamma_{2n}$ be the graph whose vertices are the $(n-1,n)$-flags of $\mathop{\mathrm{PG}}\nolimits(2n,q)$, with two vertices being adjacent if the corresponding flags are opposite. For $q$ large enough compared to $n$, a maximal coclique $\mathcal{F}$ of $\Gamma_{2n}$ falls into e

Theorems & Definitions (21)

  • Example 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.4
  • proof
  • Remark 2.6
  • Corollary 2.7
  • ...and 11 more