Results are proved indicating that the Veronese map vd often increases independence of both sets of points and sets of subspaces. For example, any d + 1 Veronesean points of degree d are independent. Similarly, the dth power map on the space of linear forms of a polynomial algebra also often increases independence of both sets of points and sets of subspaces. These ideas produce d + 1-independent families of subspaces in a natural manner
For a subset of a Lie incidence geometry two intrinsic notions of independence are introduced. Also defined is the notion of a parabolic subspace. A classification is achieved for certain independent subgraphs of the point collinearity graph of the Lie incidence geometries An,k, Bn,n, Dn,n. As a corollary it is proved that certain subspaces of these geometries are parabolic and transitivity results are obtained.