Steiner diameter, maximum degree and size of a graph

The Steiner diameter $sdia{m}_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdia{m}_2(G)=diam(G)$ 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 $\mathrm{\ell }$ was first introduced by Erd"{o}s and R'{e}nyi. Recently, Mao considered the problem of determining the minimum size of a graph of order $n$ whose Steiner $k$-diameter is at most $d$ and whose maximum is at most $\mathrm{\ell }$, where $3\le k\le n$, and studied this new problem when $k=3$. In this paper, we investigate the problem when $n-3\le k\le n$.