$3$-cluster-free families of subspaces

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 ∩ B ∩ C = \0V\ and yet (A+B+C) ≤ 2k. A special kind of $3$-cluster, which we call a covering triple, consists of subspaces $A,B,C$ such that A = (A ∩ B )⊕ (A ∩ C). We prove that, for 2 ≤ k ≤ 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.