The maximum size of a partial spread in a finite projective space

摘要

Let n and t be positive integers with t < n, and let q be a prime power. A partial (t - 1)-spread of PG(n - 1, q) is a set of (t - 1)-dimensional subspaces of PG(n - 1, q) that are pairwise disjoint. Let r = n mod t and 0 <= r < t. We prove that if t > (q(r) - 1)/(q - 1), then the maximum size, i.e., cardinality, of a partial (t - 1)-spread of PG(n - 1, q) is (q(n) - q(t+r))/(q(t) - 1) + 1. This essentially settles a main open problem in this area. Prior to this result, this maximum size was only known for r is an element of {0, 1} and for r = q = 2. (C) 2017 Elsevier Inc. All rights reserved.