The Chromatic Number of the $q$-Kneser Graph for $q \geq 5$

TL;DR

This paper determines the chromatic number of the $q$-Kneser graph $qK_{2k:k}$ for all prime powers $q \

Contribution

It provides a new result that fully characterizes the chromatic number of $q$-Kneser graphs for $q \\geq 5$, filling a significant gap in the existing literature.

Findings

Chromatic number of $qK_{2k:k}$ is determined for $q \\geq 5$

Improves previous bounds and results on intersecting families of $k$-spaces

Completes the characterization of the chromatic number for most $q$-Kneser graphs

Abstract

We obtain a new weak Hilton-Milner type result for intersecting families of $k$-spaces in $\mathbb{F}_q^{2k}$, which improves several known results. In particular the chromatic number of the $q$-Kneser graph $qK_{n:k}$ was previously known for $n > 2k$ (except for $n=2k+1$ and $q=2$) or $k < q \log q - q$. Our result determines the chromatic number of $qK_{2k:k}$ for $q \geq 5$, so that the only remaining open cases are $(n, k) = (2k, k)$ with $q \in \{ 2, 3, 4 \}$ and $(n, k) = (2k+1, k)$ with $q = 2$.