Lecture 2. Clique number
In this lecture all logs are in base . We will prove the following.
Theorem For the centered and normalized clique number
That is, as , the clique number of the Erdös-Rényi graph is .
Proof: The proof is divided into two parts. First, we show that the clique number cannot be too big using a union bound. Then we show that the clique number cannot be too small using the second moment method. In the following, denotes an Erdös-Rényi graph .
For the first part one has
Thus, choosing , we obtain
for every , where we used that for large .
For the second part it is useful to introduce some notation. Define
In particular, iff . Thus we want to show that for one has . Using the trivial observation that implies , we get
where we have used Markov’s inequality. Note that by linearity of the expectation, . Furthermore, we can write
As are boolean random variables we have and thus
which tends to for (see the first part of the proof and use the inequality ). Thus it remains to show that the following quantity tends to :
First note that, by the independence of the edges, for with we have that and are independent, so that in the numerator of the above quantity one can restrict to with . Now by an elementary reasoning we have (with being an arbitrary subset of vertices)
Clearly one has
which shows that (1) can be rewritten as
Now note that since
Using one obtains that (2) is bounded from above by
As is a convex function and as , one has . Thus for large enough the exponent in the above display is bounded from above by , and for this latter is bounded by . Thus we proved that (2) is bounded from above by
which tends to as tends to infinity, concluding the proof.