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

TL;DR

This paper determines the maximum size of a partial (t-1)-spread in finite projective spaces for a broad range of parameters, resolving a key open problem in finite geometry.

Contribution

It proves a general formula for the maximum size of partial spreads in PG(n-1,q) when t exceeds a specific bound, settling a major open question.

Findings

Maximum size formula for partial spreads established

Result applies for t > (q^r - 1)/(q - 1)

Completes the classification for most parameter cases

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 ${\rm PG}(n-1,q)$ is a set of $(t-1)$-dimensional subspaces of ${\rm PG}(n-1,q)$ that are pairwise disjoint. Let $r=n\mbox{ mod } t$ and $0\leq r<t$. We prove that if $t>(q^r-1)/(q-1)$, then the maximum size, i.e., cardinality, of a partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ 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 $r=q=2$.