Cliques in graphs with bounded minimum degree

Let $k_r(n,δ)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $δ$. We evaluate $k_r(n,δ)$ for $δ\leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture to give all extremal graphs (subject to certain conditions on $n$, $δ$ and $r$).

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