An analogue of the Erdos-Stone theorem for finite geometries

For a set G of points in PG(m-1,q), let ex(q) (G;n) denote the maximum size of a collection of points in PG(n-1,q) not containing a copy of G, up to projective equivalence. We show that lim(n ->infinity) exq(G;n)/vertical bar PG(n - 1,q)vertical bar = 1 - q(1-c,) where c is the smallest integer such that there is a rank-(m-c) flat in PG(m - 1,q) that is disjoint from G. The result is an elementary application of the density version of the Hales-Jewett Theorem.