  A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set
of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$
points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$
fixed and $q \rightarrow \infty$. In particular, we show the existence of $(3,
2)$-sets of size $(1+o(1)) q^{3/2}$ for $n=6$, $(4, 2)$-sets of size $(1+o(1))
q^{\frac{n-1}{2}}$, and $(9, 2)$-sets of size $(1+o(1)) q^2$ for $n=4$. We also
generalize a bound by Rao from 1947 and show that an $(r,s)$-set has size at
most $O(q^{\frac{n-e+1}{e}})$ if there exist integers $d,e \geq 2$ such that
$s=d(e-1)$ and $r=de-1$.
