An analogue of the Erd\H{o}s-Stone theorem for finite geometries

For a set $G$ of points in $\text{PG}(m-1,q)$, let ${\text{ex}}_q(G;n)$, denote the maximum size of a collection of points in $\text{PG}(n-1,q)$ not containing a copy of $G$, up to projective equivalence. We show that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{{\text{ex}}_q(G;n)}{\mathrm{\mid }\text{PG}(n-1,q)\mathrm{\mid }}=1-q^{1-c},$$ where $c$ is the smallest integer such that there is a rank-$(m-c)$ flat in $\text{PG}(m-1,q)$ that is disjoint from $G$. The result is an elementary application of the density version of the Hales-Jewett Theorem.