Clique-packing amplification of moment bounds ($\mathrm{Pr}(Z_k=0)\le \exp (-c{m}^2/k^8)$)

Let $X$ count $k$-cliques in a binomial random graph on $m$ vertices and $Y$ count distinct ordered pairs sharing an edge. The clique conflict graph gives $Z_k\ge X^2/(X+Y)$: order its vertices randomly and keep each one preceding all its neighbours, then apply the Cauchy-Schwarz inequality. Another application gives $\mathbb{E}{Z}_k\ge (\mathbb{E}X{)}^2/(\mathbb{E}X+\mathbb{E}Y)$. If $\mathbb{E}Y/(\mathbb{E}X{)}^2=O(k^4/m^2)$ and $1/\mathbb{E}X$ is no larger, then $\mathbb{E}{Z}_k=\Omega (m^2/k^4)$. Changing one edge changes $Z_k$ by at most one, so the Azuma-Hoeffding inequality for an edge-exposure martingale yields the displayed exponential absence bound. This supports a union bound over exponentially many adaptive candidate subsets.