  The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by
Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the
concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the
classical diameter. The problem of determining the minimum size of a graph of
order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first
introduced by Erd\"{o}s and R\'{e}nyi. In this paper, we generalize the above
problem for Steiner $k$-diameter, and study the problem of determining the
minimum size of a graph of order $n$ whose Steiner $3$-diameter is at most $d$
and whose maximum is at most $\ell$.
