On clique-to-clique densities

Let $k_r(G)$ denote the number of $r$-cliques in a graph $G$ and let $F_r(\cdot )$ be the Lovász–Simonovits $r$-clique density function. For any integers $2\le s<t$, we determine the asymptotically sharp lower bound on $k_t(G)$ in an $n$-vertex graph $G$ with a prescribed number $k_s(G)$, by showing that $$\frac{k_t(G)}{n^t}\ge F_t\text{\hspace{0.17em}⁣}\left(F_s^{-1}\text{\hspace{0.17em}⁣}\left(\frac{k_s(G)}{n^s}\right)\right),$$ where $F_s^{-1}$ denotes the generalized inverse. This strengthens Bollobás’s piecewise-linear interpolation bound and, in the case $s=2$, recovers Reiher’s clique density theorem via a new inductive proof.