Clique supersaturation by sampling ($e(G)\ge (1-1/r+ϵ)\left(\genfrac{}{}{0ex}{}{n}{2}\right)⟹k_{r+1}(G)\ge \eta n^{r+1}$)

Choose a fixed sample size $m$ for which the normalized Turan theorem bound lies below the given density by a positive margin. The expected edge count in a random $m$-vertex subset then forces a positive proportion of subsets to contain a forbidden-size clique. Each clique belongs to a known number of subsets, so double counting gives a positive constant times $n^{r+1}$ cliques.