How to prove that the probability that a random graph has a stable set of size $2\lceil \log n\rceil$ is sub-constant?
The probability that a specific set of size $k$ is a stable set is $2^{-\binom{k}{2}}$. Hence the probability that some set of size $k \leq n/2$ is stable is at most
$$
\binom{n}{k} 2^{-\binom{k}{2}} \leq 2\left(\frac{en}{k}\right)^k 2^{-k^2/2}.
$$
When $k = c\log n$ (logarithm to the base 2), this upper bound is
$$
2\left(\frac{en}{C\log n}\right)^{C\log n} 2^{-C^2\log^2 n/2}=
\frac{2^{(C-C^2/2)\log^2n}}{\Omega(\log n)^{\Omega(\log n)}} = o(2^{(C-C^2/2)\log^2n}).
$$
For $C \geq 2$, this is $o(1)$.