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

Let n and t be positive integers with t(qr−1)/(q−1), then the maximum size, i.e., cardinality, of a partial (t−1)-spread of PG(n−1,q) is (qn−qt+r)/(qt−1)+1. This essentially settles a main open problem in this area. Prior to this result, this maximum size was only known for r∈{0,1} and for r=q=2.