Note on the f-vectors of cutsets in the subspace lattice

A cutset in the subspace lattice ℒn(q) (i.e., the poset of all subspaces of an n-dimensional vector space Fqn over the finite field Fq with q elements, ordered by inclusion) is a subset of ℒn(q) that intersects every maximal chain. We find a cutset in ℒn(q) that contains a fixed percentage α (0<α≤1) of the subspaces of each possible dimension. An asymptotic estimate to the greatest lower bound of such α (denoted by α(n)) is established. In particular, limn→∞α(n)=0.