On the largest independent sets in the Kneser graph on chambers of PG(4,q)

TL;DR

This paper determines the maximum size of independent sets in a graph formed by chambers of PG(4,q) and describes their structure for large q, advancing understanding of combinatorial properties in finite projective geometries.

Contribution

It establishes the independence number of the chamber graph for q ≥ 749 and characterizes the structure of maximum independent sets.

Findings

Independence number is (q^2+q+1)(q^3+2q^2+q+1)(q+1)^2 for q ≥ 749.

Maximum independent sets have a specific geometric structure.

Provides insights into combinatorial configurations in PG(4,q).

Abstract

Let $\Gamma_4$ be the graph whose vertices are the chambers of the finite projective $4$-space PG(4,q), with two vertices being adjacent if the corresponding chambers are in general position. For $q\geq 749 $ we show that $\alpha:=(q^2+q+1)(q^3+2q^2+q+1)(q+1)^2$ is the independence number of $\Gamma_4$ and the geometric structure of independent sets with $\alpha$ vertices is described.