The Steiner 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 ℓ was first introduced by Erdös and Rényi. 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 ℓ, where 3 ≤ k ≤ n, and studied this new problem when k = 3. In this paper, we investigate the problem when n − 3 ≤ k ≤ n.