The maximum size of a partial spread II: Upper bounds

Let $n$ and $t$ be positive integers with $t<n$, and let $q$ be a prime power. A partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ is a set of $(t-1)$-dimensional subspaces of ${\rm PG}(n-1,q)$ that are pairwise disjoint. Let $r\equiv n\pmod{t}$ with $0\leq r<t$, and let ${\theta}_i=(q^i-1)/(q-1)$. We essentially prove that if $2\leq r<t\leq {\theta}_r$, then the maximum size of a partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ is bounded from above by $({\theta}_n-{\theta}_{t+r})/{\theta}_t+q^r-(q-1)(t-3)+1$. We actually give tighter bounds when certain divisibility conditions are satisfied. These bounds improve on the previously known upper bound for the maximum size partial ($t-1$)-spreads of ${\rm PG}(n-1,q)$; for instance, when $\lceil\frac{{\theta}_r}{2}\rceil+4\leq t\leq {\theta}_r$ and $q>2$. The exact value of the maximum size partial $(t-1)$-spread has been recently determined for $t>{\theta}_r$ by the authors of this paper (see N\u{a}stase-Sissokho [21]).