Maximum Erd\H{o}s-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension

TL;DR

This paper characterizes maximum independent sets in a graph formed by chambers of even-dimensional vector spaces over finite fields, confirming the structure of these sets for large q and even n.

Contribution

It proves an Erd ext{"o}s-Ko-Rado theorem for chambers in even-dimensional vector spaces over finite fields, identifying the structure of maximum independent sets.

Findings

Maximum independent sets are characterized for large q and even n.

Sets containing all chambers with a fixed 1-dimensional subspace are maximum.

The structure of these sets confirms conjectures for even n and large q.

Abstract

A chamber of the vector space $\mathbb{F}_q^n$ is a set $\{S_1,\dots,S_{n-1}\}$ of subspaces of $\mathbb{F}_q^n$ where $S_1\subset S_2\subset \dotso \subset S_{n-1}$ and $\dim(S_i)=i$ for $i=1,\dots,n-1$. By $\Gamma_n(q)$ we denote the graph whose vertices are the chambers of $\mathbb{F}_q^n$ with two chambers $C_1=\{S_1,\dots,S_{n-1}\}$ and $C_2=\{T_1,\dots,T_{n-1}\}$ adjacent in $\Gamma_n(q)$, if $S_i\cap T_{n-i}=\{0\}$ for $i=1,\dots,n-1$. The Erd\H{o}s-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of $\Gamma_n(q)$. The independence number of this graph was determined in [7] for $n$ even and given a subspace $P$ of dimension one, the set of all chambers whose subspaces of dimension $\frac n2$ contain $P$ attains the bound. The dual example of course also attains the bound. It remained open in [7] whether or not these are all maximum independent sets. Using a description from [6] of the eigenspace for the smallest eigenvalue of this graph, we prove an Erd\H{o}s-Ko-Rado theorem on chambers of $\mathbb{F}_q^n$ for sufficiently large $q$, giving an affirmative answer for n even.