Note on the -vectors of cutsets in the subspace lattice

A cutset in the subspace lattice ℒ n ( q ) (i.e., the poset of all subspaces of an n -dimensional vector space F q n over the finite field F q with q elements, ordered by inclusion) is a subset of ℒ n ( q ) that intersects every maximal chain. We find a cutset in ℒ n ( 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, lim n → ∞ α ( n ) = 0 .