Extremal sizes of subspace partitions

TL;DR

This paper determines the extremal sizes of subspace partitions in finite vector spaces, specifically the minimum and maximum sizes given constraints on subspace dimensions, and relates these to maximal partial t-spreads.

Contribution

It provides exact values for the minimum and maximum sizes of subspace partitions with given dimension constraints, extending understanding of subspace arrangements.

Findings

Exact values for _q(n,t) and _q(n,t) for all positive integers n and t.

The minimum size of a maximal partial t-spread in certain vector spaces equals _q(n,t).

Results connect subspace partitions with partial t-spreads in finite vector spaces.

Abstract

A subspace partition $\Pi$ of $V=V(n,q)$ is a collection of subspaces of $V$ such that each 1-dimensional subspace of $V$ is in exactly one subspace of $\Pi$. The size of $\Pi$ is the number of its subspaces. Let $\sigma_q(n,t)$ denote the minimum size of a subspace partition of $V$ in which the largest subspace has dimension $t$, and let $\rho_q(n,t)$ denote the maximum size of a subspace partition of $V$ in which the smallest subspace has dimension $t$. In this paper, we determine the values of $\sigma_q(n,t)$ and $\rho_q(n,t)$ for all positive integers $n$ and $t$. Furthermore, we prove that if $n\geq 2t$, then the minimum size of a maximal partial $t$-spread in $V(n+t-1,q)$ is $\sigma_q(n,t)$.