Extremal sizes of subspace partitions

A subspace partition Π 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 Π. The size of Π is the number of its subspaces. Let σq(n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρq(n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σq(n, t) and ρq(n, t) for all positive integers n and t. Furthermore, we prove that if n ≥ 2t, then the minimum size of a maximal partial t-spread in V(n + t −1, q) is σq(n, t).