Sparse clique-count concentration
= Sparse clique-count concentration
For $p=n^{-2/3}\log n$, the number $X$ of copies of $K_4$ satisfies
$$
\mathbb EX=\binom n4p^6\sim\frac{(\log n)^6}{24}
$$
and $\operatorname{var}(X)=o((\mathbb EX)^2)$. Two distinct copies have dependent indicators only when they share at least two vertices; pairs sharing two or three vertices contribute respectively $O(n^6p^{11})$ and $O(n^5p^9)$ to the variance.