  Let $S$ be a set of $n$ points in general position in the plane. The Second
Selection Lemma states that for any family of $\Theta(n^3)$ triangles spanned
by $S$, there exists a point of the plane that lies in a constant fraction of
them. For families of $\Theta(n^{3-\alpha})$ triangles, with $0\le \alpha \le
1$, there might not be a point in more than $\Theta(n^{3-2\alpha})$ of those
triangles. An empty triangle of $S$ is a triangle spanned by $S$ not containing
any point of $S$ in its interior. B\'ar\'any conjectured that there exist an
edge spanned by $S$ that is incident to a super constant number of empty
triangles of $S$. The number of empty triangles of $S$ might be $O(n^2)$; in
such a case, on average, every edge spanned by $S$ is incident to a constant
number of empty triangles. The conjecture of B\'ar\'any suggests that for the
class of empty triangles the above upper bound might not hold. In this paper we
show that, somewhat surprisingly, the above upper bound does in fact hold for
empty triangles. Specifically, we show that for any integer $n$ and real number
$0\leq \alpha \leq 1$ there exists a point set of size $n$ with
$\Theta(n^{3-\alpha})$ empty triangles such that any point of the plane is only
in $O(n^{3-2\alpha})$ empty triangles.
