$3$-cluster-free families of subspaces

Abstract

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 $A\cap B\cap C=\{0_V\}$ and yet $\dim (A+B+C)\le 2k$. 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 $2\le k\le n/2$, 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.

11 pages, 2 figures, comments welcome!