  In 1965, Paul Erd\H{o}s asked about the largest family $Y$ of $k$-sets in $\{
1, \ldots, n \}$ such that $Y$ does not contain $s+1$ pairwise disjoint sets.
This problem is commonly known as the Erd\H{o}s Matching Conjecture. We
investigate the $q$-analog of this question, that is we want to determine the
size of a largest family $Y$ of $k$-spaces in $\mathbb{F}_q^n$ such that $Y$
does not contain $s+1$ pairwise disjoint $k$-spaces. Here we call two subspaces
disjoint if they intersect trivially.
  Our main result is, slightly simplified, that if $16 s \leq \min\{
q^{\frac{n-k}{4}},$ $q^{\frac{n-2k+1}{3}} \}$, then $Y$ is either small or a
union of intersecting families. Thus we show the Erd\H{os} Matching Conjecture
for this range. The proof uses a method due to Metsch. We also discuss
constructions. In particular, we show that for larger $s$, there are large
examples which are close in size to a union of intersecting families, but
structurally different.
  As an application, we discuss the close relationship between the Erd\H{o}s
Matching Conjecture for vector spaces and Cameron-Liebler line classes (and
their generalization to $k$-spaces), a popular topic in finite geometry for the
last 30 years. More specifically, we propose the Erd\H{o}s Matching Conjecture
(for vector spaces) as an interesting variation of the classical research on
Cameron-Liebler line classes.
