  Three $k$-dimensional subspaces $A$, $B$, and $C$ of an $n$-dimensional
vector space $V$ over a finite field are called a $3$-cluster if $A \cap B \cap
C = \{\mathbf{0}_V\}$ and yet $\dim(A+B+C) \leq 2k$. A special kind of
$3$-cluster, which we call a covering triple, consists of subspaces $A,B,C$
such that $A = (A \cap B )\oplus (A \cap C)$. We prove that, for $2 \leq k \le
n/2$, the largest size of a covering triple-free family of $k$-dimensional
subspaces is the same as the size of the largest such star (a family of
subspaces all containing a designated non-zero vector). Moreover, we show that
if $k < n/2$, then stars are the only families achieving this largest size.
This in turn implies the same result for $3$-clusters, which gives the vector
space-analogue of a theorem of Mubayi for set systems.
