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 σ( n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ( 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 σ( n, t) and ρ( n, t) for all positive integers n and t. Furthermore, we prove that if n ≥ 2 t, then the minimum size of a maximal partial t-spread in V( n + t −1, q) is σ( n, t). [ABSTRACT FROM AUTHOR]