# Minimum steps adding edges to form a complete graph

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.