Almost regular partition of a graph

Let $k\le8$ be a positive integer and $G$ be a graph on $n$ vertices such that each vertex degree of $G$ is at least $\frac{k-1}kn$. It is proved in the paper that the vertex set of $G$ can be partitioned into several cliques of size at most $k$, such that for each positive integer $k_0<k$ there is at most one clique of size $k_0$ in this partition.