L. Post

Counting cliques in graphs with small independence number

We prove that for all fixed $k\geq 4$, any $N$ vertex graph with no independent set of size $n$ and $N\geq \Omega(n^{k-1}/\log^{k-2}n)$ contains at least $$ \Omega\bigg(\binom Nk \Big(\frac{\log n}{n}\Big)^{\binom k2}/\log n\bigg) $$ cliques of order $k$, and for $k\geq 5$ this is best possible conditional on the known...