Independence number of graphs with a prescribed number of cliques

We consider the following problem posed by Erdos in 1962. Suppose that $G$ is an $n$-vertex graph where the number of $s$-cliques in $G$ is $t$. How small can the independence number of $G$ be? Our main result suggests that for fixed $s$, the smallest possible independence number undergoes a transition at $t=n^{s\mathrm{/}2+o(1)}$. In the case of triangles ($s=3$) we obtain the following result which is sharp apart from constant factors and generalizes basic results in Ramsey theory: there exists $c>0$ such that every $n$-vertex graph with $t$ triangles has independence number at least $$c\cdot \min \{\sqrt{n\log n},\frac{n}{t^{1\mathrm{/}3}}{(\log \frac{n}{t^{1\mathrm{/}3}})}^{2\mathrm{/}3}\}\mathrm{.}$$