Extremal sizes of subspace partitions

A subspace partition $\mathrm{\Pi }$ 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 $\mathrm{\Pi }$. The size of $\mathrm{\Pi }$ 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 $n\ge 2t$, then the minimum size of a maximal partial $t$-spread in $V(n+t-1,q)$ is ${\sigma }_q(n,t)$.