The minimum size of a finite subspace partition

Abstract

A subspace partition of P = PG ( n , q ) is a collection of subspaces of P whose pairwise intersection is empty. Let σ q ( n , t ) denote the minimum size (i.e., minimum number of subspaces) in a subspace partition of P in which the largest subspace has dimension t. In this paper, we determine the value of σ q ( n , t ) for n ⩽ 2 t + 2 . Moreover, we use the value of σ q ( 2 t + 2 , t ) to find the minimum size of a maximal partial t-spread in PG ( 3 t + 2 , q ) .