TITLE: Minimum steps adding edges to form a complete graph
QUESTION [10 upvotes]: Consider a graph $G$ with $t$ vertices and $0$ edges. Turn it into the complete graph $K_t$ by repeatedly applying the following move $M$:
$M$: Choose $n$ vertices in $G$ and add edges between each of them to make a complete subgraph $K_n$ within $G$. This gives the new $G$.
Question: Given $t$ and $n$, what is the least number $m$ of times $M$ has to be applied before $G$ is $K_t$?

Notes:
If $n=2$ then $m$ is ${t \choose 2}$.
If $n=t-1$ then $m$ is 3.
I'd prefer a comprehensive derivation of some estimation over precise candidates for computation.

REPLY [3 votes]: No doubt you can find something much more up to date, but I happen to have a copy of a survey from 1979, so I will quote a couple of sample results from that.
The survey is: A. E. Brouwer, Packing and covering of $\binom kt$-sets, Mathematical Centre Tracts 106 (1979) 89–97.
Let $V$ be a set of $t$ vertices; the minimum size of a family $\mathcal B$ of $n$-element subsets of $V$ such that every $2$-element subset of $V$ is contained in a member of $\mathcal B$ is called the covering number $C(2,n,t).$
For $n=3,$ the following result is attributed to M. K. Fort, Jr., and G. A. Hedlund, Minimal coverings of pairs by triples, Pacific J. Math. 8 (1958) 709–719:
$$C(2,3,t)=\left\lceil\frac t3\left\lceil\frac{t-1}2\right\rceil\right\rceil\text{ for all }t.$$
For $n=4,$ the following results are attributed to W. H. Mills, On the covering of pairs by quadruples, I: J. Combin. Theory Ser. A 13 (1972) 55–78, II: J. Combin. Theory Ser. A 15 (1973) 138–166:
$$C(2,4,t)=\left\lceil\frac t4\left\lceil\frac{t-1}3\right\rceil\right\rceil\text{ for }t\ne7,9,10,19,$$
and
$$C(2,4,t)=5,8,9,31\text{ for }t=7,9,10,19.$$