I have been reading a lot on the theory of Turan graphs, but I could not find anything on the graphs I am interested in. They are somewhat between complete $k$-partite graphs and Turan graphs.
Let $G=(V,E)$ a graph such that $V$ is partitioned into $r$ subsets $V_1, \dots, V_r$. Let $n<r$ be a positive integer such that for all $1\leq m \leq n$ and for all $(i,j)\in V_m^2$, the edge $(i,j)\in E$ exists in the graph $G$. For the remainings sets, the edges do not exist in $G$: for all $n<m\leq r$ and $(i,j)\in V_m^2$, $(i,j)\notin E.$
In others words, in contrast with the Turan graphs where $G$ would contain no $(r+1)$-clique, here I consider exactly $n$ such cliques. Note that if $n=r$, then $G$ is the complete $r$-partite graph.
My question then is: what can we say about the number of edges in $G$ ?
Does this graph has a name ?
Thanks a lot!