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.