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

Abstract

Let $n$ and $t$ be positive integers with $t<n$, and let $q$ be a prime power. A $\textit{partial $(t-1)$-spread}$ of $\mathrm{PG}(n-1,q)$ is a set of $(t-1)$-dimensional subspaces of $\mathrm{PG}(n-1,q)$ that are pairwise disjoint. Let $r=n\mbox{ mod } t$ and $0\le r<t$. We prove that if $t>(q^r-1)/(q-1)$, then the maximum size, i.e., cardinality, of a partial $(t-1)$-spread of $\mathrm{PG}(n-1,q)$ 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 $r\in \{0,1\}$ and for $r=q=2$.

Journal Version (To appear in J. Combin. Theory Ser. A)