Let Vn(q) be the n-dimensional vector space over the finite field with q elements and Ln(q) denote the lattice of subspaces of Vn(q). Let , K a k-dimensional subspace and C[n,k] the set of the subspaces S such that S∩K≠=O. We show that C[n,k] is Sperner and unimodal and point out all maximum-sized antichains in C[n,k]. For the Whitney number Wm of C[n,k], we show that Wm2-qWm-1Wm+1 has nonnegative coefficients as a polynomial in q and that W0≤Wn≤W1≤Wn-1≤W2≤…. |