  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 \[\lim_{n\rightarrow \infty}
\frac{\ex_q(G;n)}{|\PG(n-1,q)|} = 1-q^{1-c},\] 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.
