TITLE: In a random graph which one is more probable, $k$-clique or $k$-core?
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: Every graph of average degree $2k$ contains a subgraph of minimum degree $k$, so the threshold for the appearance of a non-empty $k$-core is at most $2k/n$.  Since the threshold for the appearance of a $k$-clique is $n^{-2/(k-1)}$, $k$-cores typically appear before $k$-cliques.
The exact threshold for the appearance of a non-empty $k$-core was found by Pittel, Spencer and Wormald.