Recall that the *$k$-core* of a graph $G$ is the unique maximal subgraph of $G$ with minimum degree at least $k$.

In an Erdos-Renyi random graph, where the edge selection is independent with probability is $p$, we have the following inequalities: $$P(\text{$G$ contains at least one $k$-clique}) \leq \dbinom{n}{k} p^\dbinom{k}{2}$$ and $$P(\text{$G$ has a nonempty $k$-core}) \leq \dbinom{n}{m} \prod_{i=1}^m \sqrt Q,$$ where $$Q = \sum_{i=k}^{m-1} \dbinom{m-1}{i} p^i (1-p)^{m-1-i}$$ and $m$ is the size of the $k$-core.

I can't understand how I should compare these two probabilities to conclude which one is more probable.