Families of intersecting finite vector spaces

The following theorem is proved. It is a generalization of the problem for finite vector spaces analogous to a theorem of Kleitman for finite sets.Let V be an n-dimensional vector space over a finite field F of q elements. Suppose we have two collections, one consisting of k- and the other of m-dimensional subspaces of V with the property that the intersection of each member of one with each member of the other has dimension no less than r.Then if n ? k + m + 2 or n ? k + m + 1 and q ? 3, there are either no more than $\text{[}\text{n?r}\text{k?r}\text{]}{}_{\mathrm{q}}$ members in the first family or fewer than $\text{[}\text{n?r}\text{m?r}\text{]}{}_{\mathrm{q}}$ members in the second.The method used leads to a similar result for sets, provided that n ? r + (r + 1)(k ? r)(m ? r + 1) with k ? m.